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DESCARTES 


NEWTON 


GAUSS 


DURELL'S 
INTRODUCTORY  ALGEBRA 


BT 

FLETCHER  DURELL,  Ph.D. 

FEAD    OF    THE   MATHEMATICAL   DEPARTMENT   IN   THE 
LAWRENCEVILLE    SCHOOL 


NEW  YORK 
CHARLES  E.  MERRILL   COMPANY 


7-> 


n^ 


DURELL^S  MATHEMATICAL  SERIES 

ARITHMETIC 

Two  Book  Series 

Elementary  Arithmetic 

Teachers'  Edition 
Advanced  Arithmetic 
Three  Book  Series 
Book  One 
Book  Two 
Book  Three 

ALGEBRA 

Two  Book  Course 

Book  One 

Book  Two 
Introductory  Algebra 
School  Algebra 

GEOMETRY 

Plane  Geometry 
Solid  Geometry 
Plane  and  Soled  Geometry 

TRIGONOMETRY 

Plane  Trigonometry  and  Tables 
Plane  and    Spherical   Trigonometry 

AND  Tables 
Plane   and    Spherical   Trigonometry 

WITH  Surveying  and  Tables 
Logarithmic    and    Trigonometric 

Tables 


[6] 


Copyright,  1911,  1912, 
By  CHARLES  E.  MERRILL  CO. 


PREFACE 

The  main  object  in  preparing  this  new  Algebra 
has  been  to  simplify  principles  and  give  them  interest, 
by  showing  more  plainly,  if  possible,  than  has  been 
done  heretofore,  the  practical  or  common-sense  reason 
for  each  step  or  process.  For  instance,  at  the  outset  it 
is  shown  that  new  symbols  are  introduced  into  algebra 
not  arbitrarily,  but  because  of  definite  advantages 
in  representing  numbers.  Each  successive  process  is 
taken  up  for  the  sake  of  the  economy  or  new  power 
which  it  gives  as  compared  with  previous  processes. 

This  treatment  should  not  only  make  each  prin- 
ciple clearer  to  the  pupil,  but  should  give  increased 
unity  to  the  subject  as  a  whole.  We  believe  also  that 
this  treatment  of  algebra  is  better  adapted  to  the 
practical  American  spirit,  and  gives  the  study  of  the 
subject  a  larger  educational  value. 

Among  the  special  features  of  this  Introductory 
Algebra,  the  following  may  be  mentioned: 

A  large  number  of  written  problems  are  given  in  the 
early  part  of  the  book,  and  these  are  grouped  in  types 
which  correspond  in  a  measure  to  the  groups  used  in 
treating  original  exercises  in  the  author's  Geometry. 

Many  informational  facts  are  used  in  the  written 
problems.  The  central  and  permanent  numerical 
facts  in  various  departments  of  knowledge  have  been 
collected  and  tabulated  on  pages  280-286  for  use  in 
making    problems.      Similarly    the    most    important 

3 


4  PREFACE 

formulas  in  arithmetic,  geometry,  physics,  and  engi- 
neering have  been  tabulated  for  use  by  teacher  and 
pupil  (pp.  278,  279). 

The  self-activity  of  the  pupil  is  aroused  by  examples 
which  require  the  pupil  to  invent  and  solve  problems 
of  a  specified  kind,  material  for  such  examples  being 
made  available  in  the  tables  of  formulas  and  numerical 
facts. 

Many  of  the  examples  in  the  book  require  a  frequent 
review  of  the  principles  of  arithmetic,  as  of  decimal 
fractions  and  percentage. 

Numerous  and  thorough  reviews  of  the  portion  of 
the  Algebra  already  studied  are  also  called  for.  A 
unique  feature  is  the  series  of  spiral  reviews  of  the 
preceding  part  of  the  book  by  means  of  examples  at  the 
end  of  Exercises.  Oral  work  is  called  for  in  like  man- 
ner and  is  also  emphasized  in  special  important  Exer- 
cises. 

The  utilities  in  symholism  in  general,  apart  from 
technical  algebra,  are  brought  out  in  a  special  Exercise 
(pp.  249,  250)  and  thus  the  direct  practical  value  of  the 
study  of  algebra  is  much  broadened. 

The  history  of  algebra  is  discussed  in  Chapter  XIV, 
and  questions  on  this  chapter  are  inserted  in  appropri- 
ate places  in  the  text. 

The  author  wishes  to  express  his  indebtedness  to 
Professor  William  Betz  of  the  East  High  School, 
Rochester,  New  York,  and  to  Dr.  Henry  A.  Converse 
of  the  Polytechnic  Institute,  Baltimore,  Maryland, 
for  important  aid  in  preparing  the  book.  He  is  in- 
debted also  to  School  Science  and  Mathematics  and 
the  Mathematics  Teacher  for  a  few  of  the  problems. 


CONTENTS 

CHAPTER  PAGE 

I.     Algebraic  Symbols      7 

II.     Negative  Numbers 28 

III.  Addition  and  Subtraction;  the  Equation  39 

IV.  Multiplication 58 

V.     Division 76 

VI.     Equations  (Continued) 93 

VII.     Abbreviated  Multiplication  and  Division  109 

VIII.     Factoring 134 

IX.     Highest     Common    Factor    and     Lowest 

.  Common  Multiple 160 

X.     Fractions 166 

XI.     Fractional  and  Literal  Equations    ...  196 

XII.     Simultaneous  Equations 223 

XIII.     Graphs 251 

XIV.     History  of  Elementary  Algebra    ....  266 

Material  for  Examples 278 


INTRODUCTORY  ALGEBRA 

CHAPTER   I 
ALGEBRAIC   SYMBOLS 

1.  The  Use  of  Letters. 

Ex.  Walter  and  Harold  made  $27  by  gardening  one  sum- 
mer. Walter,  who  was  older  and  stronger,  received  a 
double  share  of  the  profits.     How  much  did  each  receive? 

SOLUTION   WITHOUT   THE   AID   OF  X 

1  share    =  Harold's  part  of  the  profits 

2  shares  =  Walter's  part  of  the  profits 
1  share    +  2  shares  =  $27 

3  shares  =  $27 

1  share    =    $9,  Harold's  part 

2  shares  =  $18,  Walter's  part 

SOLUTION   BY  AID   OF  X 

Let  X  —  Harold's  part  of  the  profits 

Then  2x  =  Walter's  part  of  the  profits 

Hence  a;  +  2x  =  $27 

3a;  =  $27 
X  =    $9,  Harold's  part 

2x  =  $18,  Walter's  part 

We  see  that  by  use  of  the  letter  x  the  solution  is  much 
shortened. 

2.  Algebra  is  that  branch  of  mathematics  which  treats 
of  number  by  the  extended  use  of  symbols. 

Later  algebra  comes  to  have  a  wider  meaning. 
Algebra  may  also  be  briefly  described  as  generalized  arithmetic. 

7 


8  SCHOOL  ALGEBRA 

3.  Utility  of  Algebra.  A  more  extended  use  of  symbols 
than  is  practiced  in  arithmetic  (1)  shortens  the  work  of  solv- 
ing problems;  (2)  enables  us  to  solve  problems  which  we 
could  not  otherwise  solve;  and  (3)  gives  other  advantages 
which  will  become  evident  as  we  proceed  (see  Art.  143  and 
Exercise  76,  p.  249). 

EXERCISE  1 

(Problems  of  Type  I,  i.  e.  of  the  form  x  +  ax  =  b.) 

1.  Two  boys  together  catch  84  fish.  If  the  boy  who 
owns  the  boat  which  they  use,  receives  twice  as  many  fish 
as  the  other  boy,  how  many  fish  does  each  boy  receive? 

2.  A  man  left  $12,000  to  his  son  and  daughter.  To  his 
daughter,  who  had  taken  care  of  him  in  his  old  age,  he  left  a 
double  share.    What  did  each  receive? 

3.  A  man  and  boy  by  working  a  garden  one  summer  made 
$128.80.  If  the  man  received  a  share  of  the  profits  three 
times  as  large  as  the  share  received  by  the  boy,  how  much 
did  each  receive? 

4.  -  Two  boys  together  gathered  1  bu.  4  qt.  of  hickory  nuts. 
If  the  boy  who  climbed  the  trees  received  a  double  share, 
how  many  quarts  did  each  receive? 

5.  Make  up  and  work  a  similar  example  concerning  two 
boys  who  gathered  chestnuts. 

6.  Two  girls  made  $18.60  by  sewing.  The  girl  who  sup- 
plied the  thread  and  machine  received  twice  as  much  as  the 
other  girl.    How  much  did  each  make? 

7.  Make  up  and  work  a  similar  example  concerning  two 
girls  who  kept  a  refreshment  stand. 


ALGEBRAIC  SYMBOLS  9 

8.  Solve  Ex.  1  without  the  use  of  x  (see  Art.  1).  How 
much  of  the  labor  of  writing  out  the  solution  is  saved  by  the 
use  of  xf  Is  there  any  other  advantage  in  the  use  of  x  in 
solving  a  problem? 

9.  The  total  cotton  crop  of  the  world  in  a  certain  year 
was  15,000,000  bales,  and  the  United  States  in  that  year 
produced  three  times  as  much  as  all  the  rest  of  the 
world.  How  many  bales  of  cotton  did  the  United  States 
produce? 

10.  A  farm  is  worked  on  shares.  As  the  tenant  supplied 
the  tools  and  fertilizers,  he  received  twice  as  large  a  share 
of  the  profits  as  the  owner  of  the  farm.  If  the  profits  for  one 
year  are  $6000,  how  much  does  the  tenant  receive?  The 
owner? 

11.  If  the  sum  of  the  areas  of  New  York  and  Massachu- 
setts is  57,400  sq.  mi.  approximately j  and  New  York  is  6 
times  as  large  as  Massachusetts,  what  is  the  area  of  each 
state? 

12.  One  number  is  5  times  as  large  as  another  and  the 
sum  of  the  numbers  is  240.    Find  the  numbers. 

13.  One  number  is  twice  as  large  as  another  and  the  sum 
of  the  numbers  is  7.26.    Find  the  numbers. 

14.  One  fraction  is  three  times  as  large  as  another  and  their 
sum  is  \.'  Find  the  fractions. 

15.  One  number  is  4  times  as  large  as  another  and  their 
sum  is  .0045.    Find  the  numbers. 

16.  Separate  $120  into  two  parts  such  that  one  part  is 
three  times  as  large  as  the  other. 

SuG.    Let  X  =  the  smaller  part. 


10  SCHOOL  ALGEBRA 

17.  Separate  5^^  into  two  parts  such  that  one  part  is  7 
times  as  large  as  the  other. 

18.  Make  up  and  work  an  example  similar  to  Ex.  11. 
Also  one  similar  to  Ex.  15.    To  Ex.  16. 

Material  for  examples  may  be  obtained  from  the  lists  of 
Important  Numerical  Facts  given  on  pp.  514-520. 

19.  To  look  well,  the  middle  part  of  a  steeple  should  be 
twice  as  high  as  the  lowest  part,  and  the  top  part  8  times  as 
high  as  the  lowest  part.  If  a  steeple  is  to  be  132  ft.  high, 
how  high  should  each  part  be? 

20.  A  man  wants  to  save  $6000  in  three  years.  If  he  is 
to  save  twice  as  much  the  second  year  as  the  first,  and  three 
times  as  much  the  third  year  as  the  first,  how  much  must 
he  save  each  year? 

21.  A  girl  has  $42  to  spend  for  a  hat,  coat,  and  suit.  She 
wants  to  spend  twice  as  much  for  her  coat  as  for  her  hat, 
and  three  times  as  much  for  her  suit  as  for  her  hat.  How 
much  does  she  spend  for  each? 

22.  A  man  bequeathed  $84,000  to  his  niece,  daughter,  and 
wife.  If  the  daughter  received  twice  as  much  as  the  niece, 
and  the  wife  four  times  as  much  as  the  niece,  how  much  did 
each  receive? 

23.  A  certain  kind  of  concrete  contains  twice  as  much  sand 
as  cement  and  5  times  as  much  gravel  as  cement.  How  many 
cubic  feet  of  each  of  these  materials  are  there  in  1000  cu.  yd. 
of  concrete? 

24.  Make  up  and  work  a  similar  example  for  yourself 
where  the  materials  in  the  concrete  are  as  1,  2,  4. 

25.  In  a  certain  kind  of  fertilizer  the  weight  of  the  nitrate 
of  soda  equals  that  of  the  ground  bone,  and  the  weight  of 


I 


ALGEBRAIC  SYMBOLS  11 


the  potash  is  twice  as  great  as  that  of  the  ground  bone.  How 
many  pounds  of  each  of  the  materials  are  there  in  a  ton  of 
fertiUzer? 

26.  If  the  amount  of  potash  in  a  given  kind  of  glass  is  5 
times  as  great  as  the  amount  of  lime,  and  the  amount  of 
sand  3  times  as  great  as  the  amount  of  potash,  how  many 
pounds  of  each  will  there  be  in  4000  lb.  of  glass? 

27.  The  railroad  fare  for  two  adults  and  a  boy  traveling 
for  half  fare  was  $49.50.    What  was  the  fare  for  each  person? 

SuG.    Let  X  =  the  smallest  of  the  fares. 

28.  Separate  120  into  three  parts,  such  that  the  second  part 
is  twice  as  large  as  the  first,  and  the  third  part  three  times 
as  large  as  the  first. 

29.  Separate  120  into  three  parts  which  shall  be  as  1,  2,  3. 

30.  Separate  .0372  into  three  parts  in  like  manner.  Also  y  g . 

31.  Separate  240  into  four  parts  which  shall  be  as  1,  1, 
2,4. 

32.  Separate  $1800  into  three  parts,  such  that  the  second 
is  three  times  as  large  as  the  first,  and  the  third  5  times  as 
large  as  the  second. 

33.  In  one  kind  of  concrete  the  parts  of  cement,  sand,  and 
gravel  are  as  1,  2,  and  4;  in  another  kind  three  parts  are  as 
1,  2,  and  5.  How  many  more  pounds  of  cement  are  needed 
in  a  ton  of  one  than  of  the  other? 

34.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

To  get  the  greatest  possible  benefit  from  the  use  of  letters 
to  represent  numbers,  we  now  make  further  definitions  and 
rules. 


12  SCHOOL  ALGEBRA 

4.  Three  Classes  of  Symbols.  Three  principal  kinds  of 
symbols  are  used  in  algebra:  (1)  Symbols  of  quantity,  (2) 
Symbols  of  operation,  and  (3)  Symbols  of  relation, 

5.  Symbols  for  Known  Quantities.  Known  quantities  are 
represented  in  arithmetic  by  figures;  as  2,  3,  27.  They 
are  represented  in  the  same  way  in  algebra,  but  also  in 
another  more  general  way,  viz.:  by  letters;   as  by  a,  b,  c. 

The  advantages  in  the  use  of  letters  to  represent  known  num- 
bers are:  (1)  letters  are  brief  to  write;  and  (2)  a  letter  may  stand 
for  any  known  number,  and  thus  by  the  use  of  letters  we  obtain 
results  which  are  true  for  all  numbers.     See  Exs.  34-40,  p.  97. 

6.  Symbols  for  Unknown  Quantities.  Unknown  quan- 
tities in  algebra  are  usually  denoted  by  the  last  letters  of 
the  alphabet;  as  x,  y,  z,  u,  v,  etc. 

The  advantages  in  the  use  of  distinct  symbols  for  unknown 
quantities  are  numerous  and  will  be  gradually  realized  as  we 
proceed.  Some  of  these  advantages  are  stated  in  Art.  3.  See 
also  Art.  143. 

7.  The  Signs  +,  — ,  X,  -=-,  and  =  are  used  in  algebra,  as  in 
arithmetic,  to  denote  addition,  subtraction,  multiplication, 
division,  and  equality  respectively. 

In  algebra,  multiplication  is  also  denoted  by  a  dot  placed 
between  the  two  quantities  multiplied,  or  by  placing  the 
quantities  side  by  side  without  any  intervening  symbol. 

Thus,  instead  oi  a  X  h,  we  may  write  a  &  or  ab. 

8.  Signs  of  Aggregation.  The  parenthesis  sign,  ( ),  is  used, 
as  in  arithmetic,  to  indicate  that  all  the  quantities  inclosed  by 
it  are  to  be  treated  as  a  single  quantity;  that  is,  subjected  to 
the  same  operation. 

Thus,  5(2fl  —  6  +  c)  means  that  the  quantities  inside  the  paren- 
thesis, viz.  2a,  —  b,  and  +  c,  are  each  to  be  multiplied  by  5. 


If  ALGEBRAIC  SYMBOLS  13 

kgain,  (a  +  26)  (a  +  2b  +  c)  means  that  the  sum  of  the  quanti- 
s  in  the  first  parenthesis  is  to  be  multiplied  by  the  sum  of  those 
in  the  second  parenthesis. 

Instead  of  the  parenthesis,  to  prevent  confusion,  the  fol- 
lowing signs  are  sometimes  used:  the  brackets  [  ],  the  braces 
II,  and  the  vinculum . 

9.  The  Sign  of  Continuation  is  ...  .  This  sign  is  read 
"and  so  on"  or  "and  so  on  to." 

Thus,  1,  3,  5,  7,  ...  .  is  read  ''1,  3,  5,  T  and  so  on." 
But  1,  3,  5,  7,  ...  .  19is  read  "  1,  3,  5,  7  and  so  on  to  19." 

10.  The  Sign  of  Deduction  is  .'.  and  it  is  read  "therefore" 
or  "hence." 

This  sign  is  used  to  show  the  relation  between  succeeding 
propositions. 


EXJilRCISE  2 

Express  in  words : 

1.   5  +  a. 

7. 

56  —  a. 

13. 

a  +  6  ^  3. 

2.  d  —  a. 

8. 

2a  +  3c. 

14. 

4  +  5(a  +  6). 

3.   a  ^  b. 

9. 

cd  —  ah. 

15. 

(a  +  b){x  -  y) 

4.   ad. 

10. 

7{a  +  6). 

16. 

2a  +  36  -  5c. 

5.  2a  +  36. 

11. 

7{a  -  6). 

17. 

a  ^  {x  -\-  y). 

c       d 
'-a-b 

12. 

5a  +  6 
X  +  y 

18. 

a  +  6       c 
5           d 

19.  If  a  =  1,  6  =  2,  c  =  3,   c?  =  4,  find    the   value   of 
the  combinations  of  symbols  in  Exs.  1-10. 

20.  Make  and  read  an  example  similar  to  Ex.  5.    To 
Ex.  10.    Ex.  14. 


14  SCHOOL  ALGEBRA 

Express  in  symbols: 

21.  X  plus  3.    The  sum  of  x  and  3.    The  number  which 
exceeds  a:  by  3. 

22.  X  diminished  by  3.    The  number  3  less  than  x, 

23.  Two  times  a  plus  three  times  h, 

24.  The  sum  of  4  and  of  5  times  x, 

25.  One  third  of  the  sum  of  a  and  h. 
Answer  the  following  in  algebraic  language: 

26.  If  a  boy  has  a  cents  and  earns  10  cents,  how  many 
cents  will  he  then  have? 

27.  How  many,  if  he  has  a  cents  and  earns  6  cents?    How 
many,  if  he  then  spends  c  cents? 

28.  Walter  has  x  marbles  and  his  brother  has  10  more 
than  Walter.    How  many  marbles  has  his  brother? 

29.  Walter  has  h  marbles  and  his  brother  has  5  more  than 
twice  Walter's  marbles.    How  many  has  his  brother? 

30.  If  Mary  is  a  years  old  now,  how  old  will  she  be  in  3 
years?    In  5  years?    In  x  years? 

31.  What  is  the  next  larger  number  than  5?    Than  x?    n? 
a:  +  1?    a:  +  2?    n  -  V.    x  -  21 

32.  What  is  the  next  larger  even  number  than  6?    Than 
22/?    2a:?    2n  +  2? 

33.  Taking  x  as  the  smallest  number,  write  two  consecu- 
tive numbers.    Three  consecutive  numbers.    Four.    Five. 

(The  following  problems  are  mainly  of  Type  II,  i.  e.  of 
the  form  x  +  x  +  a  =  b.) 


1  ALGEBRAIC  SYMBOLS  15 

.  If  there  are  214  pupils  in  our  school,  and  the  number 
of  girls  exceeds  the  number  of  boys  by  8,  how  many  boys  and 
how  many  girls  are  there? 

I  Let  X  =  the  number  of  boys 

Then  a;  +  8  =  the  number  of  girls 

Hence  a;  +  a:  +  8  =  214 

Or  2x  +  8  =  214 

Subtracting  8  from  the  -  8      -  8 

»  equals  gives  2x  =  206 

X  =  103,  the  number  of  boys 
X  +8  =  111 f  the  number  of  girls 

35.  Walter  and  his  brother  together  had  60  marbles,  and 
his  brother  had  10  more  than  Walter.  How  many  marbles 
had  each  boy? 

36.  Make  up  and  work  an  example  similar  to  Ex.  35. 

37.  At  New  York  on  Dec.  21,  the  night  is  5  hr.  32  min. 
longer  than  the  day.    Find  the  length  of  the  day. 

38.  Separate  28^^  into  two  parts  such  that  out.  shall  exceed 
the  other  by  2f . 

39.  A  baseball  nine  hag  played  62  games  and  won  8  more 
games  than  it  has  lost.    How  many  games  has  it  won? 

40.  In  a  certain  election  12,784  votes  were  cast.  If  the 
successful  candidate  had  a  majority  of  1732,  how  many  votes 
did  he  receive? 

41.  Make  up  and  work  an  example  similar  to  Ex.  40. 

42.  The  sum  of  two  consecutive  numbers  is  15.  Find  the 
numbers. 

43.  The  sum  of  three  consecutive  numbers  is  33.  Find 
the  numbers. 


16  SCHOOL  ALGEBRA 

44.  If  112,216  sq.  mi.  are  added  to  24  times  the  area  of 
the  British  Isles,  the  result  will  be  3,025,600  sq.  mi.  (the 
area  of  the  United  States).  Find  the  area  of  the  British 
Isles. 

45.  Twice  the  height  of  Mt.  Washington  with  1567  ft. 
added  equals  the  height  of  Pike's  Peak,  or  14,147  ft.  Find 
the  height  of  Mt.  Washington. 

46.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

47.  Which  of  the  symbols  mentioned  in  Arts.  6-10  are 
symbols  of  quantity?    Of  operation?    Of  relation? 

48.  Make  up  and  work  an  example  similar  to  Ex.  44.  To 
Ex.  45. 

Definitions  and  Principles 

11.  The  term  Factors  has  the  same  meaning  in  algebra 
as  in  arithmetic;  that  is,  the  factors  of  a  number  are  the 
numbers  which,  multiplied  together,  produce  the  given 
number. 

For  example,  the  factors  of  14  are  7  and  2;  the  factors  of  abc  are 
a,  b,  and  c. 

12.  Coefficients.  A  numerical  factor,  if  it  occurs  in  a 
product,  is  written  first  and  is  called  a  coefficient.    Hence, 

A  coefficient  is  a  number  prefixed  to  a  quantity  to  show 
how  many  times  the  given  quantity  is  taken. 

For  example,  in  5xy,  5  is  the  coefficient. 

When  the  coefficient  is  1,  the  1  is  not  written,  but  is 
understood. 

Thus,  xy  means  Ixy, 


II  DEFINITIONS  AND  PRINCIPLES  17 

The  following  enlarged  definition  of  coefficient  is  often 
used.  In  the  product  of  several  factor's,  the  coefficient  of 
any  factor,  or  factors,  is  the  product  of  the  remaining  factors. 

Thus,  in  habxy,  the  coefficient  of  y  is  5a6x;  of  xy,  is  5a6;  of  ob  is 
hxy.    What  is  the  coefficient  of  6?    Of  a?    xl    5a?    5? 

A  numerical  coefficient  is  a  coefficient  composed  only  of 
figures;  as  15  in  Ibah. 

A  literal  coefficient  is  a  coefficient  composed  only  of  letters; 
as  ah  in  ahx. 

What,  then,  is  a  mixed  coefficientf    Give  an  example  of  one. 

"  13.   Power  and  Exponent  are  used  in  the  same  sense  in  alge- 
bra as  in  arithmetic. 
Hi  A  power  is  the  product  of  equal  factors. 

^Ha.  power  is  expressed  briefly  by  the  use  of  an  exponent. 

^B/\.n  exponent  is  a  small  figure  or  letter  written  above  and 
to  the  right  of  a  quantity  to  indicate  how  many  times  the 
quantity  is  taken  as  a  factor. 

Thus,  for  xxxx,  or  four  x's  multiplied  together,  we  write  x"^,  the 
exponent  in  this  case  being  4.  The  expression  is  read  "a;  to  the 
fourth  power." 

When  the  exponent  is  unity,  it  is  omitted.  Thus,  x  is 
used  instead  of  x^,  and  means  x  to  the  first  power. 

A  power  is  composed  of  two  parts:  (1)  the  base  (i.  e.  one 
of  the  equal  factors) ;  and  (2)  the  exponent. 

Thus,  in  the  power  a',  the  base  is  a  and  the  exponent  is  3. 

14.  Root  and  Radical  Sign  have  the  same  meaning  in 
algebra  as  in  arithmetic. 

A  root  of  a  number  is  one  of  the  equal  factors  which,  when 
multipHed  together,  produce  the  given  number. 


18  SCHOOL  ALGEBRA 

The  square  root  of  a  number  is  one  of  two  equal  factors 
which,  multipHed  together,  produce  the  given  number. 
What  is  the  cube  root  of  a  number?    The  fifth  root? 

Thus,  4  is  the  cube  root  of  64,  and  a  of  a^. 

The  radical  sign  is  ^/,  and  means  that  the  root  of  the 
quantity  following  it  is  to  be  found.  The  degree  of  the  root 
is  indicated  by  a  small  figure  placed  above  the  radical  sign. 

The  number  denoting  the  degree  of  a  root  is  the  index  of 
the  root.  For  the  square  root,  the  figure  or  index  of  the  root 
is  omitted. 

Thus,  VO  means  "square  root  of  9." 
Va  means  "cube  root  of  a." 

15.  Aids  in  Solving  Problems ;  Axioms.  In  solving  prob- 
lems like  those  given  in  Exercise  1  and  the  latter  part  of 
Exercise  2,  certain  principles  are  often  important  aids  in 
discovering  the  relations  used  and  simplifying  them. 

The  most  important  of  these  principles  are  as  follows: 

1.  The  whole  is  equal  to  the  sum  of  its  parts. 

2.  Things  equal  to  the  same  things,  or  equal  things,  are  equal 
to  each  other. 

3.  //  equals  are  added  to  equals,  the  results  are  eqvxiL 

4.  //  equals  are  subtracted  from  equals,  the  results  are  equal 

5.  If  equals  are  multiplied  by  equals,  the  results  are  equal. 

6.  //  equals  are  divided  by  equals,  the  results  are  equal. 

7.  Like  powers,  or  like  roots,  of  equals  are  equal. 
These  principles  are  sometimes  called  axioms. 


ALGEBRAIC  SYMBOLS  19 

EXERCISE  3 

Write  in  words: 

1.   561  a^  +  6'  12.    Va  +  \^b. 

*•   '^  •  10.   a  +  (6  +  c)2. 

6.  2a2  4-  363.  ^^-    c         a  *  5  4  * 

16.  If  a  =  1,  6  =  2,  and  c  =  3,  find  the  value  of  the 
combinations  of  symbols  in  Exs.  1-8. 

■  17.   If  a  =  4,  6  =  8,  and  c  =  3,  find  the  value  of  the 
expressions  in  Exs.  9-12. 

Write  in  symbols : 

18.  The  square  of  the  sum  of  a  and  b.    Of  2a  minus  36. 

19.  The  cube  root  of  the  sum  of  a  and  6. 

20.  X  plus  X  increased  by  4  equals  14. 

21.  X  plus  twice  X  plus  x  increased  by  3  equals  108. 

22.  Make  up  and  work  an  example  similar  to  Ex.  18.  To 
Ex.  20. 

23.  Reduce  to  its  simplest  form  6  +  6  +  6  +  6  +  6.  Also 
6X6X6X6X6. 

If  6  =  2,  what  is  the  value  of  each  of  these  results? 

24.  Make  up  and  work  an  example  similar  to  Ex.  23. 

25.  Reduce  Saaa  +  76666  —  Bcccccc  to  its  simplest  form. 
How  many  more  symbols  are  used  in  the  long  form  than  in 
the  short  form? 


20  SCHOOL  ALGEBRA 

26.  Find  the  value  of  2"  when  n  =  1.  Also  when  n  =  2, 
3.    5.    7. 

27.  Find  the  value  of  a",  when  a  =  3  and  ?i  =  4.  Also 
when  a  =  5  and  ti  =  3. 

28.  Express  the  number  of  your  great-grandparents  as  a 
power  of  2. 

(The  following  are  miscellaneous  problems  of  Types  I 
and  II.) 

29.  A  man  and  boy  together  spade  up  a  garden  containing 
6000  sq.  ft.  If  the  man  spades  four  times  as  much  ground  as 
the  boy,  how  much  does  the  boy  spade? 

30.  Two  boys  earn  $38  by  taking  passengers  on  a  motor 
boat.  If  the  boy  who  owns  the  boat  receives  $10  more  than 
the  other  boy,  how  much  does  each  receive? 

31.  A  certain  macadam  road  cost  $1800,  of  which  the 
county  paid  twice  as  much  as  the  state,  and  the  township  the 
same  amount  as  the  county.    How  much  did  each  pay? 

32.  The  top  of  the  Statue  of  Liberty  in  New  York  Harbor 
is  306  ft.  above  the  surface  of  the  water.  If  the  altitude  of 
the  pedestal  is  4  ft.  greater  than  the  height  of  the  statue, 
how  high  is  each? 

33.  In  a  certain  kind  of  gunpowder  the  weight  of  the  char- 
coal equals  that  of  the  sulphur,  and  the  amount  of  niter  equals 
the  charcoal  and  sulphur  combined.  How  many  pounds  of 
each  substance  are  needed  to  make  a  ton  of  gunpowder? 

34.  In  a  certain  year  in  the  United  States  200,000,000 
bushels  plus  three  times  the  number  of  bushels  in  the  wheat 
crop  equaled  the  corn  crop,  or  2,600,000,000  bushels.  How 
many  bushels  were  in  the  wheat  crop? 


ALGEBRAIC  EXPRESSIONS  21 


to  Type  I.    Also  those  which  belong  to  Type  II. 
K  36.   Make  up  and  work  an  example  similar  to  Ex.  29.    To 
'"ex.  31. 

37.   How  many  of  the  examples  in  this  Exercise  can  you 

work  at  sight? 


I 


Algebraic  Expressions 


16.  An  Algebraic  Expression  is  an  algebraic  symbol  or 
combination  of  symbols  representing  some  quantity;  as 
bx^y  —  6ab  +  l\ax. 

17.  A  Term  is  a  part  of  an  algebraic  expression  which 
does  not  contain  a  plus  or  minus  sign.  (Signs  occurring 
inside  a  parenthesis  are  not  considered  in  fixing  the  terms.) 

Ex.  1.    bx^y  -  Qab  +  7\/ax. 

This  algebraic  expression  contains  three  terms:  viz.  bofy,  —  6ab, 
and  7^ ax. 

Ex.  2.    5x  +  a  -^b  -h  c. 

This  expression  also  contains  three  terms:  6x,  a  -i-  b,  and  c. 

Ex.  3.    7ax2  +  5(cf  _|.  j)  _  ^3. 

Since  the  parenthesis,  (a  +  &),  is  treated  as  a  single  quantity, 
three  terms  occur  in  this  expression:  7ax'^,  5{a  +6),  and  —  c'. 

18.  A  Monomial  is  an  algebraic  expression  of  only  one 
term;  as  5x^y  or  c. 

19.  A  Polynomial  is  an  algebraic  expression  containing 
more  than  one  term;  as  3ab  —  c  +  2a:  +  5?/^. 

A  monomial  is  sometimes  called  a  simple  expression,  and  a 
polynomial  a  compound  expression. 

20.  A  Binomial  is  an  algebraic  expression  of  two  terms; 
as  2a  —  36. 


22  SCHOOL  ALGEBRA 

A  Trinomial  is  an  algebraic  expression  of  three  terms;  as 

2a  -  36  +  5c. 

Evaluation  of  Algebraic  Expressions 

21.  The  Order  of  Operation  in  obtaining  numerical  values 
is  the  same  in  algebra  as  in  arithmetic. 

I.  In  a  series  of  operations  involving  addition,  subtrac- 
tion, multiplication,  division,  and  root  extraction,  the  multi- 
plications,  divisions,  and  root  extractions  are  to  be  performed 
before  any  of  the  additions  and  subtractions, 

Ex.  1.    Find  the  value  of  4  +  12  X  3. 

4  -K12  X  3  =  4  +  36  =  40  Ans. 
(hence  4  +12  X  3  does  not  equal  16  X  3,  etc.) 

Ex.  2.    What  is  the  value  of  60  -  (8  -^  2  +  l3  X  7? 

60  -84-2+3x7=  60  -4+ 21  =77  Ans, 

II.  If  a  given  expression  contains  one  or  more  parentheses 
(or  other  signs  of  aggregation),  each  parenthesis  is  to  be  re- 
duced  to  a  single  number  before  the  operations  of  the  expression 
as  a  whole  are  to  be  performed. 

Ex.  1.    5  +4(6  -2)  =5+4x4=5  + 16  =21  Ans. 
(hence  5  +  4(6  -  2)  does  not  equal  9(6  -  2)  or  9  X  4,  etc.) 


Note  that  in  an  expression  like  V16  +  9  the  bar  above 
the  16  +  9  is  a  vinculum,  or  sign  of  aggregation. 

Ex.  2.    Vl6  +9  =  V25  =  5  Ans. 

(hence  a/16  +  9  does  not  equal  a/16  +  V9,  etc.) 

22.  The  Numerical  Value  of  an  Algebraic  Expression  is 
obtained  thus: 

Substitute  for  each  letter  in  the  expression  the  number  which 
the  letter  stands  for; 

Perform  the  operations  indicated. 


I 


ALGEBRAIC  EXPRESSIONS  23 


I 


Thus,  if  a  =  1,  6  =  2,  c  =  3: 

Ex.  1.    Find  the  numerical  value  of  7ab  —  c\ 

7a6  -  c2  =  7  X  1  X  2  -  32 
=  14-9 
=    5  Ans. 

qh 

Ex.  2.    Find  numerical  value  of 5ah^  +  7(a'  +  2b)^  +  Sc^. 

c 


iThe  given  expression 
9  X2 


-      „      -5X1X22+  7(P  +  2  X  2)2  +  3  X  32 

=  3X2-5X4+ 7(1  +4)2  +3X9 
=  6-20  +  175+27 
=  188  Ans. 

EXERCISE  4 


In  each  of  the  following  examples,  state  the  order  of 
operations  before  working  the  example.  Wherever  possi- 
ble, use  cancellation.  When  a  =  5,  6  =  3,  c  =  1,  and 
X  =  6,  find  the  numerical  value  of 

1.  2  +  3a.  13.  a^  —  bx^. 

2.  X  -  2c.  14.  2(2a  -  c). 

3.  46  —  2x,  15.  x(a  —  6). 

4.  a  +  3a:.  16.  4(a  —  3c)^. 

5.  5a  -  Sx.  17.  2a:(2a  -  36)^. 

6.  3(a  +  c).  18.  3  +  2(a;  -  a). 

7.  a  +  3c  -  a;.  19.  5a:  -  3(26  +  c). 

8.  5a:  —  26  +  a.  20.  2(x^  —  a-)  +  3ac. 

9.  a  +  a:  -i-  6  —  c.  21.  3a:(a:  —  3)^  —  9a:. 

10.  5  x-7-  b  —  c,  22.    (a:  —  1)  (a:  —  3)  +  a:  (x—a), 

11.  36  -  x.  23.   3(2a:  -  5c)  -  a(262  -  3a:). 

12.  2a:  —  46c.  24.    (56  +  a:)  (a:  —  6  +  a  —  Sc^). 


24  SCHOOL  ALGEBRA 


25. 

a  4-  7c 

X 

26. 

3x'-b 

a-f-2c 

29 

5a2      .   3c 
X  —  1        b 

(x  -  1)  (6  +  1)  (5c  ~  b) 

2o.      ■ 


abx 


f3a^b\  (    2b    \  Uc  -  l\ 
'    \  5x  J\x-lJ\    Sb     J 


31. 

by.           33.  a'^x'^. 

38. 

6(10?/  -  36). 

39. 

3a:(4a  +  36). 

40. 

ax  +  5a!(36  -  y). 

41. 

3a  +  a(3x  -  lOy) 

42. 

5x  -  3{by  -  ab). 

If  a  =  §,  6  =  f ,  a:  =  2, 2/  =  f ,  find  the  value  of 
30.  6a.  32.  abx.  34.  3aW.  36.  2x  +  by, 

35.  X  —  26.        37.  6a6  -  Wy, 

43.  5a:  (6?/  —  a^)  —  6a:. 

44.  6(a  +  6)2  +  10(2/  -  a)2. 

45.  a:  +  VSa. 

46.  VSa  +  V36. 

47.  5?/  —  VOaa:. 

48.  Does  a:2  +  a:  =  12,  if  a:  =  2?    If  a:  =  3?    4?    5?    1? 

49.  Does  3^2  -  4a:  =  4,  if  a;  =  1?    If  a:  =  2?    3?    f  ?    0? 

50.  Doesa:2-5a:4-6  =  0,ifa:  =  1?   If  a:  =  2?   3?   4?  5? 

51.  Doesa:2-|a:-2  =  0,ifa:  =  1?    If  a:  =  2?    3?   4?^? 

52.  Show  that  (a  -  26)^  =  a^  -  4ab  +  46^,  when  a  =  3 

and  6  =  1. 

^3  _  53 

53.  That 7-  =  a^  +  a6  +  6^,  when  a  =  2  and  6  =  1. 

a  —  0 

54.  Find  the  value  of  20:^  when  a:  =  1.    When 
a:  =  2.    5.    i    1.5. 

SuG.    The  results  may  be  conveniently  arranged  as  in 
the  following  tabulation: 

Find  the  value  of  each  of  the  following  and 
tabulate  results: 

55.  2a:  +  1,  when  a:  =  1.    When  a:  =  2.    3.    5. 


X 

2a:2 

1 

2 

2 

8 

5 

50 

1 

1 

2 

2 

.5 

4.5 

1 
4 

.  1.5 

ALGEBRAIC  EXPRESSIONS  25 

16.   x^  +  2,  when  a;  =  1.    When  a;  =  2.    3.    J.    1.    5. 

M.   x{x  +  1),  when  x  =  I.    When  x  =  2.    3.    .2.    i    J. 

►8.  In  Exs.  4-10  state  which  of  the  expressions  used  are 
monomials.  Also  which  are  binomials.  Trinomials.  State 
the  same  for  Exs.  35-40. 

EXERCISE   5 

1.  If  ^  =  Iw,  find  the  value  of  A  when  ^  =  12  and  w  =  5j. 
Also  when  /  =  10.4  and  w  =  5.8. 

Do  you  know  what  use  is  made  of  the  formula  A  =  Iw  m.  arith- 
metic in  finding  areas? 

2.  If  F  =  Iwh,  find  V  when  I  =  12,  w  =  b,  and  A  =  3. 
Also  when  I  =  10.4,  w  =  5.8,  and  h  =  3.05. 

Do  you  know  what  use  is  made  of  the  formula  V  =  Iwh  in  arith- 
metic in  finding  volumes? 

^    3.   If  p  =  br,  find  p  when  b  =  350  and  r  =  1.07.     Also 
when  b  =  7.68  and  r  =  .045.    Also  when  b  =  84,000  and 

r  =  mh 

What  does  the  formula  p  =  hr  mean  in  arithmetic  in  connec- 
tion with  the  subject  of  percentage? 

'4.   If  i  =  prt,  find  i  when  p  =  $300,  r  =  .05,  and  t  =  2j. 

Also  when  p  =  $9327.50,  r  =  .06,  and  ^  =  3f . 

What  is  the  meaning  in  arithmetic  of  the  formula  i  =  prt? 

5.   If  ^  =  irR^,  find  the  value  of  A  when  tt  =  3.1416  and 
i^  =  10. 

Do  you  know  of  any  use  that  is  made  of  the  formula  A  =  ttR^ 
in  arithmetic? 


6.   If  ^  =  Va^  +  6^,  find  the  value  of  h  when  a  =  8  and 
6  =  6. 

Do   you    know  of   any    use   that   is   made    of    the    formula 
h  =  Va^  +  62  iti  arithmetic? 


26  SCHOOL  ALGEBRA 

7.  If  5  =  igfy  find  s  when  g  =  32.16  and  <  =  4.  Also 
when  g  =  32.16  and  t  =  2|. 

Can  you  find  out  the  meaning  of  the  formula  s  =  J  gf^? 

8.  A  stone  dropped  from  the  top  of  a  precipice  reaches  its 
base  in  5  seconds.    How  high  is  the  precipice? 

9.  If  C  =  UF  -  32),  find  C  when  F  =  95°.  Also  when 
F  =  100°. 

Do  you  know  the  meaning  of  the  formula  used  in  this  example? 

10.  If  iron  melts  at  a  temperature  of  2700°  F.,  at  what 
temperature  does  it  melt  on  the  centigrade  scale? 

11.  If  ^  =  itB?  -  iTr\  TT  =  3.1416,  R  =  13,  and  r  =  12, 
find  A  in  the  shortest  way. 

12.  If  1  orange  costs  3  cents,  how  many  oranges  can  be 
bought  for  12  cents?    For  x  cents?    For  x  -\-  y  cents? 

13.  If  1  orange  costs  a  cents,  how  many  oranges  can  be 
bought  for  25  cents?    For  x  cents?    For  x  -\-  y  cents? 

14.  If  1  acre  of  land  costs  x  dollars,  what  will  one  half  an 
acre  cost?    f  of  an  acre?    J  of  an  acre? 

(The  following  problems  are  variations  of  Type  I.) 

15.  If  a  12-year-old  boy  and  a  16-year-old  boy  together 
earn  $48  in  mowing  lawns,  and  the  younger  boy  receives  only 
half  as  much  as  the  other,  how  much  does  each  boy  receive? 

Let  X  =  no.  dollars  received  by  16-year-old  boy 

Then  ^x  =  no.  dollars  received  by  12-year-old  boy 

Hence  x  -\-  ^x  =  $48 

or  |x  =  $48 

Multiplying  these  equal  numbers  by  2  (Art.  15,  5) 

Sx  =  $96 
Dividing  equals  by  3  (Art.  15,  6) 

X  =  $32,  share  of  older  hoy 
\x  =  $16,  share  of  younger  boy 


K  ALGEBRAIC  EXPRESSIONS  27 

6.  A  man  left  .$24,000  to  his  son  and  daughter.  As  his 
daughter  had  cared  for  him  in  his  old  age,  he  left  his  son  only 
f  as  much  as  he  left  his  daughter.  How  much  did  each 
receive? 

17.  A  man  and  boy  together  made  $124.80  by  working  a 
garden  one  summer.  If  the  boy  received  J  as  much  as  the 
man,  how  much  did  he  receive? 

18.  A  farm  is  worked  on  shares.  As  the  owner  of  the  farm 
supplies  the  tools  and  fertilizers,  the  tenant  receives  only  f 
as  large  a  share  of  the  profits  as  the  owner.  If  the  profits  for 
one  year  are  $4410,  how  much  does  each  receive? 

19.  Two  men  manage  a  store,  and  as  one  of  them  owns 
the  building,  the  other  receives  only  f  as  large  a  share  of  the 
profits  as  the  owner  of  the  store.  If  the  profits  for  one  year 
are  $6600,  what  does  each  receive? 

20.  Separate  126  into  two  parts  such  that  one  of  them  is 
i  as  large  as  the  other,    f  as  large. 

21.  Separate  .028  in  the  same  manner  as  in  Ex.  20. 

22.  A  macadam  road  cost  $18,000.  The  county  paid  J  as 
much  of  the  cost  as  the  township,  and  the  state  paid  J  as 
much  as  the  township.    How  much  did  each  pay? 

23.  A  certain  kind  of  concrete  contained  J  as  much  sand 
as  gravel  and  J  as  much  cement  as  sand.  How  many  pounds 
of  each  material  were  there  in  If  tons  of  concrete? 

24.  Make  up  and  work  an  example  similar  to  Ex.  16.  To 
Ex.  20. 

25.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 


CHAPTER   II 
NEGATIVE  NUMBERS 

23.  Positive  and  Negative  Quantity.  Negative  quantity  is 
quantity  exactly  opposite  in  quality  or  condition  to  quantity 
taken  as  'positive. 

If  distance  east  of  a  certain  point  is  taken  as  positive,  distance 
west  of  that  point  is  called  negative. 

If  north  latitude  is  positive,  south  latitude  is  negative. 

If  temperature  above  zero  is  taken  as  positive,  temperature 
below  zero  is  negative. 

If  in  business  matters  a  man's  assets  are  his  positive  possessions, 
his  debts  are  negative  quantity. 

Positive  and  negative  quantity  are  distinguished  by  the  signs  + 
and  —  placed  before  them. 

Thus,  $50  assets  are  denoted  by  +  $50,  and  $30  debts  by  -  $30. 
We  denote  12°  above  zero  by  +  12°,  and  10°  below  zero  by  -  10°. 

The  use  of  the  signs  +  and  —  for  this  purpose,  as  well  as  to  indi- 
cate the  operations  of  addition  and  subtraction,  will  be  explained 
in  Art.  26. 

24.  Algebraic  Numbers  is  a  general  name  for  both  positive 
and  negative  numbers. 

The  absolute  value  of  a  number  is  the  value  of  the  number 
considered  without  regard  to  its  sign. 

Thus,  if  one  man  travels  5  miles  east  and  another  man  travels  5 
miles  west,  the  absolute  distance  traveled  by  the  two  men  is  the 
same,  viz.:  5  miles.  The  two  distances  traveled,  however,  are  dif- 
ferent algebraic  numbers,  one  distance  being  +  5  miles  and  the  other 
distance  being  —  5  miles. 

In  general  the  absolute  value  of  both  +  5  and  —  5  is  5;  and  of 
both  +  a  and  —  a  is  a. 

28 


I 


NEGATIVE  NUMBERS  29 


The  Utility  of  Negative  Number  lies  in  the  fact  that 
the  use  of  negative  number  enables  us  to  use  two  opposite 
or  contrasted  kinds  of  quantity  in  working  a  given  problem. 

Also  by  the  use  of  negative  quantity  we  are  often  able  to 
choose  an  advantageous  starting  point  in  solving  a  problem. 

"  The  full  meaning  of  these  utilities  and  other  advantages  in  the 
use  of  negative  quantity  will  appear  as  we  advance  in  the  study  of 
algebra. 

EXERCISE  6 

"l.  What  is  meant  by  a  temperature  of  —  8°?  By  a  latitude 
of  -  23°?  By  the  date  -  776?  (Dates  after  the  birth  of 
Christ  are  taken  as  positive.) 

2.  If  the  temperature  was  17°  at  noon  and  —  8°  at  mid- 
night, how  many  degrees  did  it  fall? 

3.  If  in  a  given  time  the  temperature  should  fall  from 
—  5°  to  —  12°,  how  many  degrees  would  it  fall? 

4.  If  the  temperature  were  15°  at  a  given  time,  what  would 
it  become  after  a  fall  of  10°?    Of  28°?    -  15°? 

5.  If  the  temperature  were  —  8°  at  a  given  time,  what 
would  it  become  after  a  rise  of  4°?    Of  15°?     -  8°? 

6.  Make  up  and  work  an  example  similar  to  Ex.  3.  Also 
to  Ex.  5. 

7.  If  a  traveler  is  in  latitude  —  4°  and  travels  north  7°, 
what  does  his  latitude  become?  What  does  it  become  if 
instead  he  travels  south  7°?  ^ 

8.  If  a  man's  property  is  —  $7000  and  he  saves  $2000  a 
year  for  8  years,  what  does  his  property  become? 

9.  If  a  vessel;  at  latitude  3°,  sails  south  345  miles,  what 
does  her  latitude  become  if  60  miles  equal  1°? 


30  SCHOOL  ALGEBRA 

10.  If  a  man  bought  a  horse  for  $L50  and  sold  it  for  $200* 
what  was  his  gain?  What  would  his  gain  have  been  if  he 
had  sold  it  for  $125?    For  $100? 

11.  What  is  meant  by  saving  —  $10?  By  a  distance  —  10 
miles  north? 

12.  What  is  the  absolute  value  of  —  4  miles?  Of  +  4 
miles?    -  5  inches?    -  3°?    -  $4200? 

13.  Make  up  an  example  for  yourself  showing  the  meaning 
of  absolute  value. 

(The  following  problems  are  variations  of  Type  II,  or  are 
of  Type  III,  viz.:  x  +  ax  +  b  =  c.) 

14.  Walter  and  his  brother  together  had  90  marbles,  and 
his  brother  had  10  less  than  Walter.    How  many  marbles  hac 
each  boy? 

Let  X  =  no.  of  marbles  Walter  had 

Then         a;  —  10  =  no.  of  marbles  his  brother  had 

X  +x  -10  =  ,90 

2a;  -  10  =  90 
Adding  10  to  each  of  these  equals  (Art.  15,  3) 

2x  =  100 

X  =  50,  no.  of  marbles  Walter  had 

x  —  10  =  40,  no.  of  marbles  his  brother  had 

15.  A  basket  ball  team  has  played  27  games  and  has  lost 
3  less  than  it  has  won.    How  many  games  has  it  won? 

16.  In  a  certain  election  12,420  votes  were  cast,  and  the 
defeated  candidate  had  210  less  votes  than  the  winning  can- 
didate.   How  many  votes  had  each  candidate? 

17.  Make  up  and  work  a  similar  example  for  yourself. 

18.  Walter  and  his  brother  together  have  83  marbles.  If 
his  brother  has  7  less  than  twice  the  number  Walter  has, 
how  many  has  each  boy? 


NEGATIVE  NUMBERS  31 

19.  One  number  exceeds  4  times  another  number  by  5, 
and  the  sum  of  the  numbers  is  100.    Find  the  numbers. 

20.  One  number  exceeds  3  times  another  number  by  .12, 
and  the  sum  of  the  numbers  is  4.4.    Find  the  numbers. 

21.  One  fraction  exceeds  5  times  another  fraction  by  ^, 
and  the  sum  of  the  fractions  is  V  •    Find  the  fractions. 

22.  The  distance  from  New  York  to  Chicago  is  912  miles. 
If  this  is  24  miles  less  than  ^  times  the  distance  from  New 
York  to  Boston,  what  is  the  latter  distance? 

23.  The  Eiffel  Tower  is  984  ft.  high.  If  this  is  126  ft.  less 
than  twice  the  height  of  the  Washington  Monument,  what 
is  the  height  of  the  Washington  Monument? 

24.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

25.  Which  of  Exs.  14-23  are  of  type  x  -{-  x  —  a  =  b,  and 
which  are  of  type  x  -\-  ax  ^  h  =  c'l 

26.  Make  up  and  work  an  example  similar  to  Ex.  19.  To 
Ex.  23. 

26.  Double  Use  of  +  and  —  Signs.  The  signs  +  and  — 
are  employed  for  two  purposes  (see  Arts.  7  and  23) :  first, 
to  indicate  the  operations  of  addition  and  subtraction;  and 
second,  to  express  positive  and  negative  quantity.  We  are 
able  to  make  this  double  use  of  these  signs  because,  in  each 
use,  the  signs  are  governed  by  the  same  laws. 

--7   -6   -5  -4  -3   -2  -1      .     +1   +2  +3  +4    +5  +6  +7  +8 

w I     I     I     I     I     I     I     I     I     I  ^ 

K         B  O         A  F  -^ 

A  person  walks  from  0  toward  E  a  distance  of  5  miles  (to  F)  and 
then  walks  back  toward  W  a  distance  of  3  miles  (to  A).    If  the  dis- 


32  SCHOOL  ALGEBRA 

tance  to  the  right  of  0  is  regarded  as  positive,  and  therefore  the  dis- 
tance to  the  left  of  0  is  negative,  the  distance  from  the  starting 
point  to  the  destination  may  be  expressed  as  the  sum  of  a  positive 
quantity  and  a  negative  quantity;  that  is, 

(positive  distance  OF)  +  (negative  distance  FA)^ 

or,  +5  +  (-3)  =  5  -3  =  2. 

The  position  arrived  at  may  be  determined  in  another  waj?"  —  viz. 
by  deducting  3  miles  from  5  miles.    We  obtain 

5  -  (+3)  =  5  -3  =  2. 

From  this  example  we  see  that  adding  negative  quantity  is 
the  same  in  effect  as  subtracting  'positive  quantity. 

Therefore,  in  the  expression  5  —  3,  the  minus  sign  may 
be  considered  either  a  sign  of  the  quality  of  3,  or  as  a  sign  of 
operation  to  be  performed  on  3.  Hence,  we  are  able  to  use 
the  signs  +  and  —  to  cover  two  meanings. 

27.  Laws  for  the  Use  of  +  and  —  Signs.  Whichever  of 
the  two  meanings  of  +  and  —  named  in  Art.  26  is  assigned, 
we  see  that  +  (-  3)  =  -  3;   also,  -  (-h  3)  =  -  3. 

The  signs  +  and  —  applied  in  succession  to  a  quantity  are 
equivalent  to  the  single  sign  — . 

Or  in  symbols, 

4-  (—  a)  =  —  a;  and  _  (+  a)  =  —  a. 

Ex.    Find  the  value  of  8  +  4-11  +  3- 6.    On  squared 
paper  show  the  meaning  of  the  numbers  involved. 
8+4- 11  +3-6  =  15 -17=  -2  Ans. 

Taking  the  distances  to  the  right  of  OP  as  positive,  we  have  the 
diagram  on  p.  33  showing  the  meaning  of  the  numbers  involved. 

Note  that  the  above  process  holds  true  whether  a  number  pre- 
ceded by  a  minus  sign  is  regarded  as  the  subtraction  of  a  positive 
number  or  the  addition  of  a  negative  number. 

If  in  the  illustration  on  p.  31  a  person  walks  in  the  nega- 
tive direction  from  0  (i.  e.  toward  W)  a  distance  of  4  miles 


I 


NEGATIVE  NUMBERS 


33 


to  K,  and  then  reverses  his  direction  and  goes  2  miles,  he 
will  be  at  B.      Or  stated  in  another  way,  diminishing  the 


n 

+ 

8 

+ 

4 

-11 

+  3 

- 

2 

—^ 

_— 

^^ 

'6 

F 

distance  traveled  west  by  2  miles,  brings  him  to  the  same 
place  as  walking  the  full  direction  west  and  then  walking  2 
miles  east. 

It  may  be  well  to  study  another  illustration  of  this  principle. 
If  a  man  owes  two  notes  of  $500  and  $100  respectively,  removing 
the  note  for  $100  is  the  same  in  effect  as  annexing  $100  in  money  to 
the  debts  as  they  are.    That  is, 

-  $500  -  $100  -  (-  $100)  =  -  $500  ^  $100  +  $100  =  -  $500 

Hence: 

The  sign  —  applied  twice  to  a  given  positive  quantity  gives 
a  +  result. 

Or  in  symbols,  —  (—  a)  =  -f  a. 

These  laws  enable  us  to  u^e 
negative  quantity  with  as  great 
freedom  as  we  u^e  positive  quan- 
tity, and  hence  are  an  important 
source  of  power,  as  will  become 
more  evident  later. 

Ex.    On  squared    paper  show 
the  meaning  of    —  5  —  (—  3).    Also  of  —5  +  3.    Hence, 
show  that  -  5  -  (-  3)  =  -  5  +  3. 


4 

~5 

n^ 

D 

1 

r 

i-3 

\ 

\ 

34  SCHOOL  ALGEBRA 

On  the  lower  diagram  on  p.  33  —  5  —  ( —  3)  means  OA  —  DA, 
.or  OD.     Also  -5+3  means  OA  +  BC,  or  OD. 

Hence,  —  5  —  (—  3)  and  —5+3  give  the  same  result;  or  we 
may  say  — 5  —  (—  3)  =  — 5+3. 

28.  The  Algebraic  Sum  of  two  or  more  algebraic  numbers 
is  the  result  of  combining  the  given  algebraic  numbers  into 
a  single  number. 

Thus,  the  algebraic  sum  of  4  and  —  7  is  —  3. 

EXERCISE  7 

Find  the  value  of  each  of  the  following  and  verify  the 
result  on  squared  paper: 

1.  5  -  2.  6.  0  -  4.  11.  5  -  (-  8). 

2.  6  -  8.  7.  8  -  6  —4. .  12.  -  7  +  (-  2). 

3.  5  -  5.  8.  7  -  5  +  4.  13.  0  -  (-  5). 
4.-4  +  2.  9.  3  +  1  -  5.  14.  0  +  (-  5). 
5.-4-2.  10.  -  4  -  (-  3).  15.  -  4  -  (-  1.5). 

16.  4  +  5  -  12  +  3  -  5. 

17.  -3  +  8-6-2  +  2-1. 

18.  At  6  A.  M.  a  thermometer  read  57°.  It  then  made 
successive  changes  as  follows:  +  7°,  —  2°,  +  5°,  —  3°,  —  2°. 
What  was  the  final  reading? 

19.  In  a  certain  football  game,  taking  a  distance  toward 
the  north  goal  as  positive,  during  the  first  seven  plays  the  ball 
started  at  the  middle  of  the  field  and  shifted  its  position 
in  yards  as  follows:  +50-10-15-5  +  10-5-20. 
Find  the  final  position  of  the  ball  with  reference  to  the  middle 
of  the  fitild.  On  squared  paper  show  the  changes  in  the  posi- 
tion of  the  ball,  letting  5  yd.  equal  one  space  on  the  papePr   ; 


NEGATIVE  NUMBERS  35 

20.  State  in  the  language  of  debts  and  credits  the  mean- 
ing of 

-  $700  -  $200  -  (  -  $200)  =  -  $700  -  $200  +  $200 

!Hi    SuG.    If  a  man  has  debts  of  $700  and  $200,  the  removal  of  the 
$200  debt  is  the  same  as  leaving  his  debts  unchanged  and  adding 
i_S200  to  his  possessions.    He  becomes  worth  —  $700  in  either  case. 

^^  21.   State  in  the  language  of  distance  traveled  east  and 
west  the  meaning  of 
-  10  mi.  -  2  mi.  -  (  -  2  mi.)  =  -  10  mi.  -  2  mi.  +  2  mi. 

(The  following  are  miscellaneous  problems  of  Type  II  and 
Type  III.) 

22.  A  man  and  a  boy  together  catch  320  fish,  and  the  man 
receives  three  times  as  many  fish  as  the  boy.  How  many  fish 
does  each  have? 

23.  A  man  has  $3220  in  two  banks  and  the  amount  in  one 
bank  exceeds  that  in  the  other  by  $540.  How  much  has  he 
in  each  bank? 

,  24.  Two  girls  make  $24.60  by  sewing,  and  the  younger 
girl  receives  only  one  half  as  much  as  the  older.  How  much 
does  each  receive? 

25.  Separate  $12.68  into  two  parts  one  of  which  shall  be 
smaller  than  the  other  by  $5. 

26.  A  given  piece  of  bronze  weighs  4600  lb.  It  contains 
twice  as  much  tin  as  zinc,  and  8J  times  as  much  copper  as 
zinc.  How  many  pounds  of  each  metal  does  the  bronze 
contain? 

27.  The  distance  from  the  mouth  of  the  Mississippi  River 
to  the  source  of  the  Missouri  River  is  4500  miles.  The  dis- 
tance between  the  mouth  of  the  Mississippi  and  the  mouth  of 


36  SCHOOL  ALGEBRA 

the  Missouri  is  1700  miles  less  than  the  length  of  the  Missouri. 
What  is  the  length  of  the  Missouri? 

28.  A  farmer  obtained  2720  pounds  of  cream  in  one  month 
by  the  use  of  a  separator.  This  is  ^  more  than  he  would 
have  obtained  if  his  milk  had  been  skimmed  by  hand.  How 
much  would  he  have  obtained  by  the  latter  process? 

29.  The  cost  of  a  macadam  road  was  $24,000.  The  county 
paid  twice  as  much  as  the  state,  and  the  township  three  times 
as  much  as  the  state.    How  much  did  each  pay? 

30.  Three  partners  divided  $14,000,  the  second  partner 
receiving  $2000  more  than  the  first,  and  the  third  partner  re- 
ceiving twice  as  much  as  the  first.  How  much  did  each 
receive? 

31.  Mt.  Washington  is  6290  ft.  high.  This  is  170  ft.  more 
than  10  times  the  height  of  the  Singer  Building  (N.  Y.). 
How  high  is  the  latter? 

32.  Make  up  and  work  an  example  similar  to  Ex.  18. 
Ex.21.    Ex.24.    Ex.29. 

33.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

34.  Which  of  Exs.  22-33  of  this  Exercise  are  of  Type  I? 
Of  Type  n?    TypeHI? 

29.  Graphs.  A  set  of  numerical  facts  may  often  be  com- 
bined as  a  geometrical  picture  called  a  graph.  The  meaning 
and  use  of  negative  numbers  are  often  well  illustrated  on  a 
graph. 

Ex.  On  a  given  day  the  following  were  the  temperatures 
at  a  given  place: 


NEGATIVE   NUMBERS 


37 


Midnight  -  15°      9  A.  M. 
3  A.  M.     -  20°      Noon 
6  A.  M.     -  10°      3  P.  M. 


2° 
10° 
15° 


6  P.  M.         10° 
9  P.  M.  0° 

Midnight  -  10° 


Temperatures 
20' 


Hours 


Graph  these  facts. 

We  draw  a  horizon- 
tal line  and  on  it  mark 
oflf  spaces  to  represent 
hours,  as  in  the  dia- 
gram. Perpendicular  to 
this  we  draw  a  line  and 
on  it  mark  off  spaces  to 
represent  temperatures. 
Above  or  below  each 
point  which  represents 
an  hour,  a  point  is  lo- 
cated which  represents 
the  temperature  at  that 
hour.  Through  the  points  thus  located  a  continuous  line  ABCD 
is  drawn.    This  is  the  required  graph. 

EXERCISE  8 
Graph  each  of  the  following  sets  of  temperatures: 


1. 

Mid- 
night 

3A.M. 

6A.M. 

9  A.M. 

12  M. 

3  P.M. 

6  P.M. 

9  P.M. 

-20° 

-30° 

-20° 

-10° 

0° 

10° 

10° 

0° 

2. 

-10° 

-20° 

-10° 

0° 

10° 

20° 

10° 

6° 

3. 

-10° 

-15° 

-  5° 

10° 

15° 

25° 

15° 

-5° 

4. 

0° 

-10° 

-  5° 

15° 

25° 

30° 

15° 

5° 

5.  Make  up  and  work  a  similar  example  for  yourself. 
Graph  each  of  the  following  sets  of  temperatures: 


6. 
7. 
8. 


Jan.  1 

Feb.  1 

Mar.  1 

Apr.  1. 

May  1 

June  1 

New  York 
New  York 
London 

31° 

-1°C. 
37° 

31° 

-1°C. 
38° 

35° 

1°C. 
40° 

42° 

6°C. 
45° 

54° 
12°  C. 
50° 

64° 
18°  C. 

57° 

38 


SCHOOL  ALGEBRA 


July  1 

Aug.  1 

Sept.  1 

Oct.  1 

Nov.  1 

Dec.  1 

71° 
22°  C. 
62° 

73° 
23°  C. 

62° 

69° 
21°  C. 
59° 

61° 
16°  C. 
54° 

49° 

9°C. 
46° 

39° 
4°C. 
41° 

9.  Convert  the  temperatures  given  for  London  in  Ex.  8 
to  temperatures  on  the  Centigrade  scale  and  graph  them 
(see  Ex.  9,  p.  26). 

10.  Collect  and  graph  sets  of  numerical  facts  similar  to 
those  given  in  the  preceding  examples. 


H    ADDITION  AND  SUBTRACTION;  THE  EQUATION 

^K  Addition 

30.  The  Utility  of  Addition  in  Algebra. 

Ex.     Find  the  value  of  3a6^  +  Saft^  +  2a}? y  when  a  =  2 


f 


nd  6  =  3. 


PROCESS   WITHOUT  ALGEBRAIC   ADDITION 


If  we  substitute  directly  in  the  given  expression,  we  obtain 
3a62  +  5a62  +  2a62  =3x2x32+5x2x32+2x2x32 

=  54+90+36 

=  180  Ans. 

PROCESS  AIDED   BY  ALGEBRAIC   ADDITION 

3a62  +  5a62  +  2ah''  =  lOafe^ 

=  10  X  2  X  32 
=  180  Am, 

In  solving  the  above  example,  algebraic  addition  enables 
us  to  save  more  than  half  the  work.  Algebraic  addition  has 
other  uses  which  will  appear  later. 

Why  do  we  now  make  definitions  and  rules? 

31.  Addition,  in  algebra,  is  the  combination  of  several 
algebraic  expressions  into  a  single  equivalent  expression. 

Addition  is  sometimes  described  as  collecting  terms  in  an  expression. 

32.  Similar  Terms  (or  like  terms)  are  terms  which  contain 
the  same  literal  factors  and  the  same  radical  signs  over  the 
same  factors. 

•  39 


40  SCHOOL  ALGEBRA 

Thus,  lab"^  and  —  bab"^  are  similar  terms.  Also  5a  V  2  and  —  6a  V2 
are  similar  terms. 

Dissimilar  terms  (or  unlike  terms)  are  terms  which  are 
unlike  either  in  their  literal  factors  or  in  the  radical  sign 
over  the  same  factor. 

Thus,  Sa^ft  and  bab"^  are  dissimilar  terms.  Also  3v  5  and  3a/5 
are  dissimilar  terms. 

The  addition  of  dissimilar  terms  can  only  be  indicated. 

Thus,  h  added  to  a  gives  a  +  h;  also  o?  —  Sa%  added  to  Sa^  —  ¥ 
gives  a3  -  3a%  +  Sa^  -  bK 

33.   Method  for  Addition.    The  most  convenient  general 
method  for  addition  is  shown  in  the  following  examples: 
Ex.  1.   Add  4a:2  +  3a!  +  2,  3x^  -  4x  -  3,  -  2x^  -  x  -  5. 

Arranging  similar  terms  in  the  same  column,  and  adding  each 
column  separately,  we  obtain 

CHECK 

4x2+ 3a; +2=  4+3+2=       9 

3x^  -4x  -3   =  3  -4  -3  =  -4 

-2x2-    X  -  5   =    -2-1-5^ 8 

Sum  5x2  -2x -6=  5-2-6=-3 

To  check  the  accuracy  of  the  work,  we  let  x  =  any  convenient 
number,  as  ,1;  find  the  numerical  value  of  each  row;  and  compare 
the  sum  of  these  results  with  the  numerical  value  of  the  algebraic 
expression  obtained  as  the  sum. 

Ex.  2.  Add  2a^  -  5a^h  +  Aa¥  +  aW,  ^o?h  +  2o^  -  a¥  -  3ah\ 
a^h  -a'  +  2ah\ 

Proceeding  as  in  Ex.  1, 

CHECK 

2a3  -  5a26  +  4a62  +  a^h^             =2-5+4  +  1  =2 

2a3  +  4a26  -  ZalP-             -aft^  =  2+4-3-1  =2 

-  a^  +    a26  +  2a62 =      -1+1+2  =2 

Sum  3a3              +  3a62  +  ^253  _  fj^^   =  3+0+3  +  I-I  =6 

In  the  second  column   the  algebraic  sum  of  the  coefficients  is 
—  5+4  +  1,  which  =  0;  and  as  zero  times  a  number  is  zero,  the 


I 


ADDITION  41 


sum  of  the  second  column  is  zero,  which  need  not  be  set  down  in 
the  result. 

The  work  is  checked  by  letting  a  and  h  each  =  1. 

Hence,  the  process  for  addition  may  be  stated  as  follows: 

^jk^rraiige  the  terms  to  be  added  in  columns,  placing  similar 
terirns  in  the  same  column; 

Find  the  algebraic  sum  of  the  numerical  coefficients  jof  each 
column  and  prefix  this  result  to  the  literal  factors  common  to  the 
terms  in  the  column. 

Sometimes  the  algebraic  sum  of  the  coefficients  of  each  group  of 
similar  terms  is  found  without  arranging  the  terms  in  columns. 


I 


EXERCISE  9 
dd  and  check  each  result: 


1. 

2. 

3. 

4. 

5. 

- 11 

4 

Sx 

—    x 

-7x 

6 

-  10 

-  6x 

-  3x 

12x 

6. 

7. 

8. 

9. 

10. 

2a 

-x" 

Ixy 

a'b 

7xV 

ba 

3x2 

-  10x1/ 

Ba'b 

-  lOxV 

-  12a 

5x2 

2xy 

-Sa^b 

xY 

11.  3ax,  —  2ax,  5ax,  ax,  —  3ax. 

12.  5x2,  i2a:2,  -  lOx^,  x^,  -  IGx^,  Sx^,  -  x\ 

13.  7aW,  -  \2aW,  -  aW,  -  ^aW,  baW,  Mb\ 


14. 

15. 

16. 

17. 

3(a  +  b) 

-  6(x  -  y) 

b^a-\-x 

47rr2 

5(a  +  b) 

^{x  -  y) 

-  eVa  4-  X 

-27rr2 

-  4(a  +  b) 

-  5(x  -  y) 

2Va  +  x 

i-r^ 

42  SCHOOL  ALGEBRA 


18. 

19. 

20. 

3a; -22/ 

5^2+    7 

a^  —    ax  -\-  Ax^ 

-  2x  +  31/ 

x^  -  10 

3a2  -f  2aa;  -  Sar^ 

X-    y 

-7a;2+    1 

—  0?  —    ax  —    x^ 

21.  a  —  26,  3a  +  46,  a  +  56,  —  5a  —  6,  a  —  56. 

22.  ,3a:2  +  2/2,  2a:2  -  T?/^,  -  4ir2  _  5^2^  ^^  _|_  3^2^  _  3^2^ 

23.  3aa:^  —  56?/^,  2aa:^  +  46i/^,  26?/^  —  4aa:^,  6^/^  —  aa:^. 
Reduce  each  of  the  following  to  its  simplest  form: 

24.  x"  -  xy  +  Sy^  +  2^2  +  2xy-2y^  +  x^  +  y^  +  3x^  -xy. 

25.  mn  —  3ri2-j-  772,2  -|-  m^  +  2^2  —  Smn  +  m2  —  7i2  -^  jjifi  _  2^2. 

26.  a:2  +  2/2  -  2^2  +  3^2  -  2/2  +  222  +  2;2  -  2a:2  +  a:2  _  ^2. 

27.  2a:2  _  a:^/  +  3xy  -  5y^  +  32^2  _  3^^  _^  ^.-2  +  2y^  -  2xy. 

28.  7x  +  y  -{-  5z  —  lOxy  -\-  2y  —  3z  +  ISxy  —  Axz  +  52 

—  6x  —  4:xz  -\-2xy  —  ^y  -{-  9z  -{-  7x  —  xz  -{-  2\xz  —  I62  -\-x  —  5xy. 

29.  0^  +  ^x'^y  +  3a:2/2  _[_  ^^  _|-  ^^3  _  3^2^  _j_  3^^2  —  y^  -\-  2x^y 

—  2xy'^  -\-y^  +  x^  —  y^  +  x'^y  —  4:0^  —  xy^  —  i^  +  y^  +  a^  — 
x^y  +  xy"^. 

Collect  similar  terms  in  the  following  and  check  each  result: 

30.  2a;  -  3?/  -  5a:  +  42  +  4i/  +  z  - 2?/  -  a:  -  32  +2.T-3y. 

31.  Zxy  —  5ax  +  3y^  —  2xy  —  3a;2  +  4aa;  —  2y^  +  3aa;  —  2xy. 

32.  a;  -  3i/  +  22  +  2^/  -  2a;  -  2  -  3a;  -  42-2a;  +  2  +  2a;. 

33.  2a;  -  1  +  5?/  -  2  +  3a;  +  2  +  32/  -  3  -  2.T  + 1  -  a;  -  32/. 

34.  3d26  -  2a2c  +  3a2  -  5a26  -  a2  -  3a2c  +  a26  +  6a2c  -  2a2. 

35.  5a^  -Sx  +  4:-2x^  -63^  +  Ax-7  -  x'^  +  a^  -i-  3x^ 

—  a;  -I-  5  +  3a;2  -  6a;  -  a;2  +  4a;  -  2a;2  +  2a;. 

36.  2a;"  -  5a:"*  +  3a;2  -  a;"  -  7a;  +  3a;2  -  3  +  2x"'  -  bx^ 
+  5  +  3a;"*. 


I 


ADDITION  43 


7.  Reduce  Sxxxyy  +  Sxxxyy  —  5xxxyy  —  2xxxyy  to  its 
simplest  form.  About  how  much  briefer  is  the  form  you 
obtain  than  the  given  form? 

38.  Make  up  and  work  an  example  similar  to  Ex.  37. 

39.  Make  up  and  work  an  example  showing  the  use  of 
algebraic  addition  (see  Ex.  of  Art.  30,  p.  39). 

40.  State  in  general  language  the  use  of  algebraic  addition. 
|m(The  following  are  mixed  problems  of  Types  I,  II,  III.) 

41.  Three  partners  in  a  retail  business  made  $18,000  in 
one  year.  The  second  partner  owned  the  building  and  re- 
ceived twice  as  large  a  share  as  the  first  partner.  The  third 
partner  supplied  most  of  the  capital  and  received  three  times 
as  large  a  share  as  the  first.    How  much  did  each  receive? 

42.  Make  up  and  work  a  similar  example  for  yourself. 

43.  Find  three  consecutive  numbers  whose  sum  is  36. 

44.  Find  four  consecutive  numbers  whose  sum  is  106. 

45.  Make  up  and  work  an  example  concerning  five  con- 
secutive numbers. 

46.  The  area  of  the  United  States  and  its  outlying  posses- 
sions is  3,742,155  sq.  mi.  The  area  of  the  United  States 
exceeds  that  of  its  outlying  possessions  by  2,309,045  sq.  mi. 
What  is  the  area  of  the  outlying  possessions? 

47.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

48.  Name  the  type  to  which  each  of  the  above  problems 
belongs  (Exs.  41-46). 


44  SCHOOL  ALGEBRA 

Subtraction 

34.  The  Utility  of  Subtraction  in  Algebra. 

Ex.  Find  the  numerical  value  of  17aW  —  15aW,  wheq 
a  =  3  and  6  =  2. 

PROCESS  WITHOUT  ALGEBRAIC  SUBTRACTION 

17a263  _  I5a%^  =  17  X  3^  x  2^  -  15  X  3^  x  2' 
=  17X9   X8   -15X9X8 
=  1224  -  1080  =  144  Am. 

PROCESS  AIDED   BY  ALGEBRAIC   SUBTRACTION 

=  2  X  32  X  23 
=  144  Ans. 

In  solving  the  above  example,  algebraic  subtraction 
enables  us  to  save  more  than  half  of  the  work.  Algebraic 
subtraction  has  other  advantages  which  will  appear  later. 

Why  do  we  now  proceed  to  make  definitions  and  rules? 

35.  Subtraction,  in  algebra,  is  the  process  of  finding  a 
quantity  which,  added  to  a  given  quantity  (the  subtrahend), 
will  produce  another  given  quantity  (the  minuend). 

Thus,  if  we  subtract  Sab  from  lOab,  we  obtain  7ab,  for  7ab 
added  to  Sab  (subtrahend)  gives  lOab  (minuend). 

36.  Signs  in  Subtraction.    From  Art.  26  it  follows  that 
Subtracting  a  positive  quantity  is  the  same  as  adding  a  nega- 

tive  quantity  of  the  same  absolute  magnitude;  and 

Subtracting  a  negative  quantity  is  the  same  as  adding  a 
positive  quantity  of  the  same  absolute  magnitude. 

37.  Method  for  Subtraction.  The  most  convenient  general 
method  in  subtraction  is  to 


I 


SUBTRACTION  45 


CHECK 

5x'  -2x^  +    X  -S   =5 

-2+1-3=1 

-  2x3  a  3a,2  _    3.  4.  2   =2 

-3-1+2=0 

Write  the  terms  of  the  subtrahend  under  the  terms  of  the 
minuend,  placing  similar  terms  in  the  same  column; 

Change  the  signs  of  the  terms  in  the  subtrahend  mentally,  and 
proceed  as  in  addition. 

Ex.  1.  From  5a^-2x^ -j- x  -  3  subtract  2a^  -  3x'^  -  x  +  2. 
Cheek  the  work  by  letting  x  =  1. 

I_..„............. 

|^P?he  coefficient  of  a;^  is  5  -  2,  or  3,  of  a:^  is  -  2  +  3,  or  1,  etc. 

Ex.  2.  Subtract  2a'  -  3a'b  -  6a''b^  -  2a¥  +  2b'  from 
a'  +  5a^6  —  Qa^b^  —  3a¥.  Check  the  work  by  letting  a  =  1 
and  6  =  1. 

CHECK 

^4  +  5^35  _  Qa%^  _  ^a¥  =       1+5-6-3  =-3 

-  2a^  +  Sa%  -  Qa%^  -  2ab^  ^2¥  =       2-3-6-2+2=  -7 

-  a^  +  8a^b  -a¥-2b^=-l-\-S-i-0-l-2=      4 

The  coefficient  of  a'^¥  is  -  6  +  6,  or  0.    The  coefficient  of  b^  is 
0  -  2,  or  -  2. 


EXERCISE  10 

Subtract  and  check  each  result : 

1. 

2.                3. 

4.                    5. 

6. 

7ab 

5ic             X 

5a:           -  3a:2 

-  7a:2/ 

Sab 

9a;            2a: 

-  3a:          -  4a:2 

Sxy 

7. 

8. 

7(a:  +  y) 

9. 

10. 

5(a  +  b) 

-  2Va  +  X       - 

-  4V6  _  2/ 

3(a  +  6) 

-  3(a:  +  y) 

-  sVa  +  a: 

2Vb-y 

11. 

12. 

13. 

14. 

3x^  -  4x 

3a: -9 

2x?  -b 

5a:2  +  4a:  -  3 

2x^+    X 

5a: +  1 

-  x?^2 

ar^  -  3a:  +  5 

46  SCHOOL  ALGEBRA 

15.  From  3a  +  2b  -  3c  -  d  take  2a  -  2b  +  c  -  2d, 

16.  From  7  -  3a:  +  2x'^  take  15  -  4x  -  5x^. 

17.  From  ic^  -  2/2  -  2;2  +  8  take  2x''  +  y^  -  2z^  +  10. 

18.  From  bxy  —  3xz  +  ^yz  +  x^  take  ^xz  —  20*2/  —  ^r^. 

19.  From  2  -  x  +  x^  -{-x"^  take  3  +  x-a:2-a:3_  2x\ 

20.  Subtract  lOa:^?/  +  3x'^y'^  —  \3xy'^  from  x^?/  —  xy"^  +  2a;2y2^ 

21.  Subtract  3  —  2ab  +  3ac  —  Acd  from  5  —  ac  +  Scd  —  5a^. 

22.  Subtract  1  +  x  —  x'^  +  o^  —  x^  from  2  —  a:  —  a:^  —  a:^.^^^ 

23.  Subtract  a  -\-  2b  —  3c  -{-  4d  from  m  +  26  +  c?  —  a:  +a. 

24.  Subtract  3a;^  -  2a:2  _^  5^  _  7  ^j.^^^  Sa:^  ^  2a:2  -  a:  -  7. 

25.  Subtract  -  a:^  -  2a;^  +  a:^  +  5  from  a:^  -  a:^  +  a:^  -  2a:  +  5. 

26.  Subtract  3a:'"  —  3x"  +  a:  —  3  from  a:"»  +  a:"  —  a:^  +  a:  —  1. 

27.  From  the  sum  of  2a:  and  3y  subtract  their  difference. 

28.  From    0    subtract  —  3a:.     From  0  subtract  x  —  y. 
From  zero  subtract  3a2  —  2ab  +  6^. 

29.  Reduce  laaabb  +  baaahb  —  3aaabb  to  its  simplest 
form. 

30.  Make  up  and  work  an  example  similar  to  Ex.  29. 

If  ^  =  a:^  -  3a:2  +  1,  5  =  2a:2  -  5a:  -  3,  C  =  3a:3  +  a:2  +  3^^ 
find  the  value  of 

31.  A  +  B  +  C  33.  A  +  B  -  C 

32.  B  -  A-\-  C  3i.  A  -  B  +  C 

(The  following  problems  are  variations  of  Types  I  and  II.) 

35.  Find  the  value  of  a:,  if  3a:  —  2  in.  =  7  in. 

36.  Separate  $24.80  into  two  parts  such  that  one  part  is 
smaller  than  the  other  by  $4.60. 


I 


USE  OF  THE  PARENTHESIS  47 


37.  Separate  $24.80  into  two  pai-ts  such  that  the  smaller 
!part  equals  |  of  the  larger  part. 

I  38.  Separate  $5000  into  three  parts  such  that  the  second 
part  shall  exceed  the  first  by  $300,  and  the  third  shall  exceed 
the  first  by  $800. 

39.  Separate  $5000  into  three  parts  such  that  the  second 
part  shall  exceed  the  first  by  $300,  and  the  third  shall  exceed 
the  second  by  $800. 

40.  Separate  $6000  into  three  parts  such  that  the  second 
part  equals  J  of  the  first,  and  the  third  part  equals  ^  of  the 
first. 

41.  Separate  $6000  into  three  parts  such  that  the  second 
part  is  double  the  first,  and  the  third  part  is  double  the  second. 

42.  Make  up  and  work  an  example  similar  to  Ex.  36. 
To  Ex.  40. 

43.  Name  the  type  of  which  each  of  the  above  problems 
(Exs.  36-43)  is  a  variation. 

44.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

45.  How  many  of  the  examples  in  Exercise  1  can  you  now 
work  at  sight? 

Use  of  the  Parenthesis 

38.  Utility  of  the  Parenthesis.  The  parenthesis  is  useful 
in  indicating  an  addition  or  a  subtraction  in  a  brief  way. 

Thus,  2a  +  36  -  5c  -  (3a  -  26  +  3c)  indicates  that  3a  -  26  +  3c 
is  to  be  subtracted  from  2a  +  36  —  5c. 

The  parenthesis  will  also  be  found  useful  in  indicating 
multiplication  and  division  in  a  brief  manner,  and  other  uses 
of  the  parenthesis  will  become  evident  as  we  proceed 


48  SCHOOL  ALGEBRA 

39.  Removal  of  a  Parenthesis.  From  the  processes  of  addi- 
tion and  subtraction  it  follows  that 

When  a  parenthesis  preceded  by  a  -\-  sign  is  removed,  the 
signs  of  the  terms  inclosed  by  the  parenthesis  remain  unchanged. 

But 

When  a  parenthesis  preceded  by  a  minus  sign  is  removed,  the 
signs  of  the  terms  inclosed  by  the  parenthesis  are  changed,  the 
+  signs  to  —,  and  the  —  signs  to  -\-. 

Ex.    Simplify  2a  +  36  -  5c  -  (3a  -  26  +  3c). 

2a  +  36  -  5c  -  (3a  -  26  +  3c)  =  2a  +  36  -  5c  -  3a  +  26  -  3c 

=  —  a  +  56  —  8c  Ans. 
Let  the  pupil  check  the  work  by  letting  a  =  l,6  =  l,c  =  l. 

40.  Parenthesis  within  Parenthesis.  Using  the  parenthesis 
as  a  general  name  for  all  the  signs  of  aggregation,  it  is  evident 
that  several  parentheses  may  occur  one  within  another  in 
the  same  algebraic  expression.  The  best  general  method  of 
removing  several  parentheses  occurring  thus  is  as  follows : 

Remove  the  parentheses  one  at  a  time,  beginning  with  the 
{innermost; 

Collect  the  terms  of  the  result. 

It  is  also  possible  to  remove  the  parentheses  in  reverse  order, 
that  is,  by  removing  the  outside  parenthesis  first,  etc.  Working  an 
example  in  this  way  often  forms  a  convenient  check  on  the  first 
process. 


I 


Ex.    Simplify  bx 

-  y  - 

[4a:  -  6?/  +  {  - 

3a: +  2/ -f- 

&x  -  z)}]. 

5x  -    y  - 

-[ix- 

-62/  +  1 

-  3x  +  ?/  +  22  - 

(2a:  -2)11 

=  5x  -    y  - 

-[4x- 

-6y  +  { 

-Zx  +y  +2z  - 

2x  +  2}] 

=  5x  -    y  - 

-[4a:- 

-62/ 

-Zx  +y  ■\-2z  - 

2a:  +2] 

=  5x  -    y  - 

-      4:X 

+  6?/ 

+  3x  -  2/  -  22  4 

-   2x  -z 

=  6x  +^y  - 

-  Sz  Ans. 

USE  OF  THE  PARENTHESIS  49 

The  work  may  be  checked  by  removing  the  parentheses  in  reverse 
order,  or  by  the  method  of  substitution  as  follows: 

Letting  x  =  1,  y  =  2^  z  =  S,  we  have 

5x  -y  -l^x  -Qy  +  {  -  Sx  -\- y  +2z  -  {2x  -  z)}] 
=  5-2- [4- 12  {  -3+2+6- (2-  3)}] 
=  3-[-8  +  {5-(-  1)|]  =  3  -  [  -  8  +  {5  +  1}] 
=  3-(-8+6)  =3-(-2)  =3+2=5 
Also  6a; +42/  -32  =6+8-9=5 

EXERCISE  11 

Remove  parentheses  and  collect  similar  terms.  Check 
each  result  either  by  substitution  of  numerical  values,  or  by 
reversing  the  order  in  which  the  parentheses  are  removed. 

.    1.  3a  +  (2a  -  6).  7.   x  -  [2x  -\-  (x  -  1)]. 

2.  2x-  (x-  1).  8.   5a:  +  (1  -  [2  -  40!]). 

3.  a:  +  (1  -  2a:).  9.    2  -  11  -  (3  -  a)  -  a}. 

4.  3a;  -  (1  +  3a;).  lo.   2a:  -  [-  a:  -  (a;  -  1)]. 

5.  X-  (-X-  1).  11.   2y+  {-X-  (2y  -  x)}. 

6.  X  +  2y  -  {2x  —  y).  12.   a  -  {-  a  —  (—  a  —  1)^. 

13.  [a:^  -  (x^y  -  z^)  -  z^]  +  (x'^y  -  x^). 

14.  1  -  {1  -  [1  +  (1  -  a:)  -  1]  -  1}  -a:. 

15.  a:  -[-{-(-  a:  -  1)  -  a:}  -  1]  -  1. 

16.  1  -  12  +  [-  3  -  (-  4  -  5-6)  -  7]^ 

17.  a-  \a-h[b  -  (a+  6  +  c-a  +  6  +  cZ)-  c]}. 

18.  X  -  {2x^  +  (3a^  -  3a:  -  [a:  +  x^])  +  [2a:  -  (x^  +  a^)]}, 

19.  x'  -  [4a:3  _  [3^.2  _  ^2x  +  2)]  +  3a:]  -  [x'  +  (Sa^  -\-2x^ 
-  3a:  -  1)]. 

20.  -  [-  2a:  -  {-.  (-  2a:  -  1)  -  2a:i  -  1]  -  2a:. 

21.  X  -[x  +  (a:  -  2/)  -  {^  +  (2/  -  ^)  -  22/}  +  2/]  -  2/  +  ^. 


1 


50  SCHOOL  ALGEBRA 


22.  25a:  -  [12  +  ^3a:  -  7  -  (-  12a:  -  5  +  15a:)  -  (3  -f 
2a:)}]  +  7  -  (3a:  +  5)  +  (2a:  -  3)  +  a:  +  8. 

23.  In  3a:  —  (5a  —  26  +  c),  what  is  the  sign  of  5a  as  the 
example  stands? 

24.  In  Ex.  12,  Exercise  10,  indicate  the  subtraction  by 
use  of  a  parenthesis.    Do  the  same  in  Ex.  13. 

Remove  the  parentheses  and  find  the  value  of  x  in  each 
of  the  following: 

25.  X+  {x  +  2)  =  7.  27.   5a:  -  (2a:  -  3)  =  12. 

26.  3a:  -  (a:  +  2)  =  8.        28.    4a:  -  (a:  -  |)  =  2j. 

29.  Make  up  and  work  an  example  similar  to  Ex.  16.  To 
Ex.  26. 

30.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

31.  How  many  of  the  examples  in  Exercise  3  (p.  19)  can 
you  work  at  sight? 

41.  Insertion  of  a  Parenthesis.  It  is  clear  that  the  process 
of  removing  a  parenthesis  may  be  reversed;  that  is,  that 
terms  may  be  inclosed  in  a  parenthesis. 

Inverting  the  statements  of  Art.  39,  we  have 

Terms  may  he  inclosed  in  a  parenthesis  preceded  by  the  plus 
sign,  provided  the  signs  of  the  terms  remain  unchanged; 

Terms  may  be  inclosed  in  a  parenthesis  preceded  by  the  minus 
sign,  provided  the  signs  of  the  terms  are  changed. 

Ex.    a— b-\-c-\-d— e  =  a  —  b  +  (c  -{-  d  —  e), 

or,  =a— b  —  {— c— d-\-e)  Ans. 


USE  OF  THE  PARENTHESIS  51 

EXERCISE    12 

In  each  of  the  following  insert  a  parenthesis  inclosing  the 
last  three  terms,  each  parenthesis  to  be  preceded  by  a  minus 
sign.  Check  the  work  either  by  removing  the  parenthesis  in 
the  answer,  or  by  numerical  substitution. 

1.  0^3  -  3a:2  _|_  3^  _  1^       5    ^4  _|_  4^  _  ^2  _  ^ 

2.  a-b  +  c-\-d.  6.   a262  -  2cd  -  c^  -  (P. 

3.  1  +  2a  -  a2  -  1.  7.   ^x^  -  9a:2  +  12xy  -  Ay\ 

4.  1  -  a2  -  2ab  -  h\        q.   a:^  -  4a^  +  42:^  +  4a:  -  4  -  r^. 

It  is  often  useful  to  collect  the  coefficients  of  a  letter  into 
a  single  coefficient. 

Ex.    Collect  the  coefficients  of  x,  y,  and  z  in  the  expression, 
3x  —  4y  -^  5z  —  ax  —  by  —  cz  —  hx  -\-  ay  -i-  az. 

The  complete  coefficient  of  a:  is  (3  —a  —  6) ;  of  2/,  ( —  4  —  b  +  a) 
or  —  (4  H-  6  —  a) ;  of  2;,  (5  —  c  -}-  a). 
Hence,  the  expression  may  be  written, 

(3  —  a  —  b)x  —  (4  +  6  —  a)y  +  (5  —  c  +  a)z  Ans. 

In  like  manner  collect  the  coefficients  of  x,  y,  and  z: 

9.   mx  —  ny  -{-  3z  +  2x  -{-  nz  —  4cy. 

10.  X  —  y  —  2z  —  ax  -\-  by  —  az  —  bx  —  ay  -\-  cz. 

11.  -  7x  +  12y  -  lOz  -  2ax  +  36^  -  cy  -{-  2bx  -  6dy . 

12.  ^y  —  Sacx  —  5cdz  —  4abx  —  3cdy  +  2cx  —  4:Z  —  bax. 
Collect  the  coefficients  of  :x?,  x^,  and  x\ 

13.  ?>d?  -\-  X  —  2x^  —  ao:^  —  5  +  aa:^  —  2ax  —  c:i?  —  cx^  —ex, 

14.  —  x^  —  x  —  ax^  -\-  :>?  —  ax  -\-bx^  —  ax?  —  3bx  —  2ba^-\-  3a, 

15.  a^x^  —  Qx  —  a  —  Wx?  —  26V  +  Zbx  —  aV  —  cx^  -{- 
3cx  —  c. 


52  SCHOOL  ALGEBRA 

Equations  and  Transposition 

42.  An  Equation  is  a  statement  of  the  equality  of  two 
algebraic  expressions. 

An  equation,  therefore,  consists  of  the  sign  of  equality  and 
an  algebraic  expression  on  each  side  of  it;  as  3a:  —  1  =  2a;  +  5. 

The  solution  of  an  equation  is  the  process  of  finding  the 
value  of  the  unknown  number  (as  of  x)  in  the  given  equation. 

43.  Members  of  an  Equation.  The  algebraic  expression 
to  the  left  of  the  sign  of  equality  is  called  the  first  member  of 
the  equation;  the  expression  to  the  right  of  the  sign  of  equality 
is  called  the  second  member. 

Thus,  in  the  equation  3x  —  1  =  2a;  +  3, 
the  first  member  is  3a;  —  1;  the  second  member  is  2x  +  3. 

The  members  of  an  equation  are  sometimes  called  sides  of 
the  equation. 

The  members  of  an  equation  are  similar  to  the  pans  of  a  set  of 
weighing  scales  which  must  be  kept  balanced.    (See  Art.  15,  p.  18.) 

44.  Utility  of  Equations.  An  equation  expresses  the  re- 
lation of  at  least  one  unknown  quantity  to  certain  known 
quantities.  By  means  of  an  equation,  we  are  often  able  to 
determine  the  value  of  the  unknown  quantity. 

See  the  problems  solved  in  Exercises  1,  2,  etc.,  by  the  aid  of 
equations. 

I      45.   The  Transposition  of  a  Term  is  moving  the  term  from 
one  member  of  an  equation  to  the  other  member.    We  shall 
see  that  when  a  term  is  transposed,  the  sign  of  the  term  must 
be  changed. 
Ex.  1.    Find  the  value  of  a;  in  x— 5  =  7. 


I 


i 


r 

EQUATIONS 

i 

PROCESS 

WITHOUT  TRANSPOSITION 

We  have 

given 

x-5  = 
5  = 

7 
5 

53 


I 


Adding  5  to  each  of  the  equals,        x  =  7  +  5 

or  a;  =  12  Ans. 


(Art.  15,  3) 


PROCESS   WITH  TRANSPOSITION 


We  have  given  x—5=    7 
Transferring  5  to  the  right-hand^  x  =    7+5 

member    of    the    equation    and^  a;  =  12  Ans. 

changing  its  sign,  J 


Hence  transposition  is  a  short  way  of  adding  equal  num- 
bers to  the  two  members  of  an  equation.  The  labor  saved 
by  means  of  transposition  is  more  evident  when  several  terms 
are  to  be  transposed  at  the  same  time. 

For  the  present,  however,  in  order  to  fix  firmly  in  mind  the  na- 
ture of  the  process,  we  shall  not  transpose  terms,  but  shall  add 
equals  to  the  members  of  an  equation  when  we  wish  to  transfer 
terms  from  one  member  to  the  other. 

Ex.  2.    Solve  5x-(x  +  2)  =  3x  -  {2x  -  7).  //-^  V 

Removing  parentheses,  5x—x  —  2  =  3a;— 2a;  +  7 

Adding  -  3x  +  2x  +  2  to  j         _  3;^  +  2x  +  2  =  -  3x  +  2x  +  2 
each  member,  ) 

5a;  -  a;  -  3a;  +  2a;  =  2  +  7 
3a;  =  9 
a;  =  3  Ans. 

46.  Checking  the  Solution  of  an  Equation.  The  result 
obtained  by  solving  an  equation  may  be  checked  by  substi- 
tuting in  each  member  of  the  original  equation  the  value  of 
X  obtained  by  the  solution.  If  the  two  members  reduce  to 
the  same  number,  the  value  found  for  x  is  correct. 

Thus,  in  Ex.  2,  putting  3  in  the  place  of  x, 
The  left  member,  5a;  -  (a;  +  2)  =  15  -  (3  +  2)  =  15  -  5  =  10 
Also  the  right  member,  3a;  -  (2a;  -  7)  =  9  -  (6  -  7)  =9  +  1  =10 


54  SCHOOL  ALGEBRA 

EXERCISE   13 

Solve  the  following  equations  without  transposition  o{ 
terms.    Verify  each  result  obtained. 


1. 

o:  +  2x  -  3  =  6. 

6.    3a:  -  2  -  2a:  +  .74. 

2. 

3a:  =  a:  +  10. 

7.   5a:  -  (2a:  -  3)  =  6. 

3. 

5a:  -  1  =  14. 

8.   7a:  -  {5x  +  4)  =  -2. 

4. 

4a:  -  3  =  12  -  a:. 

9.   9a:  =  10  -  (a:  +  5). 

5. 

5a:  -  1  =  3a:  +  7. 

10.   8a:  +  (3a:  -  4)  =  25. 

11.    10a:  -  (x  ■ 

-  5)  =  4  -  (a:  +  2). 

12.    10  -  (3a: 

-  5)  =  8  -  (7:c  +  2). 

13.  Solve  Exs.  1-12  by  aid  of  transposition  of  terms. 
Solve  the  following  problems  and  cheek  each  result: 

14.  If  5  times  x  equals  9  diminished  by  twice  a:,  find  x, 

15.  If  f  of  a:  equals  12  less  Ja;,  find  x. 

16.  If  12  is  added  to  a  given  number,  the  result  equals 
three  times  the  given  number.    Find  the  number. 

17.  One  number  exceeds  another  by  5  and  the  sum  of  the 
numbers  is  12.    Find  the  numbers. 

18.  The  difference  of  two  numbers  is  5  and  the  sum  of  the 
numbers  is  13.    Find  the  numbers. 

19.  Separate  12  into  two  parts  such  that  one  part  exceeds 
the  other  by  5. 

20.  One  number  exceeds  another  by  1.4  and  the  sum  of 
the  numbers  is  16.4.    Find  the  numbers. 

21.  The  difference  of  two  numbers  is  1.4  and  the  sum  of 
the  numbers  is  16.4.    Find  the  numbers. 


EQUATIONS  55 

22.  Separate  16.4  into  two  parts  such  that  one  part  ex- 
ceeds the  other  by  1.4. 

23.  Make  up  and  work  three  examples  similar  to  Exs. 
14-16.  'Also  to  Exs.  17-20. 

24.  Find  three  consecutive  odd  numbers  whose  sum  is  45. 
Also  five  consecutive  odd  numbers  whose  sum  is  45. 

25.  Find  three  consecutive  even  numbers  whose  sum  is  60. 
Also  five  consecutive  even  numbers  whose  sum  is  60. 

26.  Make  up  and  work  an  example  similar  to  Ex.  24. 

27.  Make  up  and  work  an  example  similar  to  Ex.  25. 

28.  To  what  type  does  each  of  the  above  problems  belong, 
or  of  what  type  is  each  a  variation? 

EXERCISE   14 

Review 

1.  Find  the  value  of  a  +  3(6  —  a;),  when  a  =  5,  6  =  2,  and 
x=  I. 

2.  Find    the    value    of    Sx  -  {x  —  2)2  +  2(a;  +  1)  (4  -  a:)  - 
V^x  +  1,  when  x  =  S. 

3.  li  s  =  vt  +  igt^,  find  the  value  of  s  when  v  =  10,  g  =  32.16, 
and  t  =  4. 

4.  lix  =  3,  find  the  value  of  ix'^.    Also  of  {^xY. 
Simplify: 

5.  )2x*  -^5a^  -  3a;2  +  2x  -  5  -\- 2oi^  -  Sx*  -2x  +  2x^  -  2x  +  2x^ 
-6  +,3.t2  +  *4  -^x^  _[.  7_X_f-  2  +  3a:'  +  2a;^  -  4x  -2x^. 

6.  3a/2~-  5Vs  +  8  +6^-  2v^ -  7  +  3V3"-  4VT-  2. 
Subtract: 

7.  3x3  _2x^  -^5x  -^  from  8x^  -  x^  -  1. 

8.  5x3  -  3^.2^  +  yz  from  Sx^  +  7xy^  -  y\ 


56  SCHOOL  ALGEBRA 

Simplify  and  collect: 
9.   Sx  -  {  -  2x  +  [-  4x  -  {x  -  2)  -  x]  -  x}  -  1. 

10.  9x-{-8x-[7x  +  i-ex  +  l)  -dx]-  4a;}  -(3a:  +  1)  -  2x. 
Bracket  coefficients  of  like  powers  of  x: 

11.  x^  -x^+2-Sx*-  ax^  +  ax^  -cx^-  2ax^  +  3cx^  -  2cx^-5x\ 

12.  1  -  X  -  x"^  -  x^  +2a  -  2ax  +  2ax^  -  2ax^  -  3bx  +  Sbx"^ 
+  Sbx^  +  ex. 

Solve  and  give  the  reason  for  each  step : 
•  13.    3a;  -  5  =  X  +  7.  15.  4a;  +  (a;  -  1)  =  3a;  -  (x  +2). 

14.   5a;  -  (a;  -  4)  =  16.  16.  3  -  (a;  -  2)  =  7  -  5a;. 

17.  Subtract  5a;2  -  3aa;  -  2a^  from  -  3a;2  +  2ax^  -  a\ 

18.  Find  the  value  of  5x^  —  3  (a  —  2a;)  +  Sa^,  when  a  =  4  and 
a;  =  L 

19.  Add  5a;2  —  3aa;  +  4a2,  5aa;  —  3a;2  +  a^,  and  3aa;  —  x^  —  2ax. 

20.  Simphfy  a;^  -  [5aa;  +  {a^  -  2x^  -  ax)  -  3x^\  -  5a2.  Test  the 
accuracy  of  your  work  by  letting  a  =  1  and  a;  =  2. 

21.  Solve  5-a;=4-(7+  3a;). 

22.  The  land  surface  of  the  world  is  51,240,000  square  miles.  If 
the  land  area  of  the  rest  of  the  world  is  seven  times  that  of  North 
America,  find  the  area  of  North  America. 

23.  Add  ia;2  -  |a;+  i,  \x^  +  ia;  -  J,  and  \x^  -Ja;  +  f. 

24.  Subtract  \x^  —  \x-\-\  from  \x'^-  \x  -  f . 

25.  Add  .5a2-.15a  +2.5,  1.2a2  +.3a-1.5,  and-.75a2  +  .3a-.7. 

26.  Subtract  .27a2  -  .12a  -  2.3  from  l.Sa^  +  2a  -  1.7. 

27.  Add  2(a;  +  2/)  -  3(a;+  z)  +  2fo  +  z),  4(a;  +  2)  -  3(a;  +  y) 
-  5(2/  +  z),  and  4(a;  +  ?/)  -  (a;  +  2)  +  4  (?/  +  z). 

28.  From  the  sum  of  a^  -  7ah  +  36^  and  2a2  -  662  +  7a'^b^,  take 
the  sum  of  4:a%^  -  3a^  +  2a2  -  b^  and  3a6  -  2¥  +  a^. 

29.  What  must  be  added  to  x"  —  a;  + 1  that  the  sum  may  be  a;^? 
That  the  sum  may  be  3a;?    15?    0? 

30.  What  must  be  subtracted  from  2a;2  —  3a;  +  1  that  the  re- 
mainder may  be  a;^?    x^  +  10?    7?    a  -  a;  +  1? 


EQUATIONS  57 


^P    B  =4x^  -    x^y  -    xy^  -  Zy\        D  =    x^  -  2xy^  +    y\ 
find  the  value  of 

31.  A  -  5  +  C  -  D  33.   A  -  (5  +  C)  +  Z) 

32.  A  -[B  -{D  -{■  C)]  34.   B  +  \A  -[C  -  D]] 

35.  By  a  diagram  show  that  —  7  —  ( —  3)  and  —7+3  have 
the  same  value. 

36.  In  an  election  for  two  candidates,  32,544  votes  were  cast. 
The  successful  candidate  had  a  majority  of  2416  votes.  How  many 
votes  did  each  candidate  receive? 

37.  The  Panama  Canal  is  49  miles  long  and  the  part  of  it  through 
the  lowlands  is  4  miles  more  than  8  times  the  part  through  the  hills 
(called  the  Culebra  Cut).    How  long  is  each  part? 

38.  How  many  examples  in  Exercise  2  (p.  13)  can  you  now  work 

at  sight? 


CHAPTER   IV 

MULTIPLICATION 

47.  Multiplication,  at  the  outset,  may  be  regarded  as  the 
process  of  finding  the  result  (called  the  product)  of  taking  one 
quantity  (the  multiplicand)  as  many  times  as  there  are  units 
in  another  quantity  (the  multiplier). 

The  term  multiplication  has  acquired  a  much  broader 
meaning  than  this,  which  is  sometimes  expressed  as  follows: 

Multiplication  is  the  process  of  finding  a  number  (the 
product)  which  is  obtained  from  a  given  number  (the  multi- 
plicand) in  the  same  way  that  another  number  (the  multiplier) 
is  obtained  from  unity. 

Multiplication  is  useful  as  a  means  of  shortening  addition 
or  subtraction.  Later  many  other  uses  (often  indirect)  of 
multiplication  will  become  evident. 

Multiplication  of  Monomials 

48.  Multiplication  of  CoeflBicients.  To  multiply  4a  by  36, 
we  evidently  take  the  product  of  all  the  factors  of  the 
multiplier  and  the  multiplicand,  and  thus  get  4  X  a  X  3  X  6. 
Rearranging  factors,  we  obtain  as  the  product, 

4X3XaX6or  12ab. 

Hence,  in  multiplying  two  monomials. 

Multiply  the  coefficients  to  prodvxie  the  coefficient  of  the 
product, 

58 


MULTIPLICATION  OF  MONOMIALS  59 

49.  Multiplication  of  Literal  Factors  or  Law  of  Exponents. 
Ex.    Multiply  a^  by  a\ 

Since  a^  =  a  X  a  X  a 
and  a^  =  a  X  a 
: .  a?  X  a^  =  a  X  a  X  a  X  a  X  a  '=  a\ 

This  may  be  expressed  in  the  form 

a^  Xa?  =  a^'''^  =  a^, 
or,  in  general,  a"*  X  a"  =  a"*  "*"  ", 

;vhere  m  and  n  are  positive  whole  numbers. 

Hence,  in  multiplying  the  literal  factors  of  a  monomial, 

Add  the  exponents  of  each  letter  that  occurs  in  both  multiplier 
ind  multiplicand. 

Ex.    4a26c3  X  U^h^x  =  l2aW(^x. 

50.  Law  of  Signs.  The  law  of  signs  in  multiplication 
"ollows  directly  from  the  general  law  of  signs  as  stated  in 
irt.  31. 

(1)  +  $100  taken  5  times  gives  +  $500, 

)r,  in  general,  a  +  quantity  taken  a   +  number  of  times  gives 

a  +  result. 

(2)  $100  of  debts,  that  is,  -  $100,  taken  5  times  gives  -  $500, 
)r,  in  general,  a  —  quantity  taken  a  +  number  of  times,  gives 

a  —  quantity  as  a  result. 

(3)  $100  deducted  5  times,  or  $100  X  -  5,  gives  as  the  total 

amount  of  deduction  —  $500, 

)r,  in  general,  a  +  quantity  taken  a   —  number  of  times,  gives 

a  —  quantity  as  a  result. 

(4)  Deducting  $100  of  debts  5  times  from  a  man's  possessions 

is  the  same  as  adding   $500   to   his   assets;   that  is, 
-  $100  X  -  5  =  +  $500, 
ir,  in  general,  a  —  quantity  taken  a   —  number  of  times  gives 
a  +  quantity  as  a  result. 


60  SCHOOL  ALGEBRA 

We  see  from  (1)  and  (4)  that 

either  +  X  +,  or  —  X  — ,  gives  +, 
and  from  (2)  and  (3),  that 

either  —  X  +,  or  +  X  — ,  gives  — . 
In  brief,  in  multipHcation 
lAhe  signs  give  plus;  unlike  signs  give  minu^. 

51.  Multiplication  of  Monomials.  Combining  the  results 
of  Arts.  48,  49,  and  50,  we  may  express  the  method  of  multi- 
plying one  monomial  by  another  as  follows : 

Multiply  the  coefficients  together  for  a  new  coefficient; 

Annex  the  literal  factors,  giving  each  factor  an  exponent 
equal  to  the  sum  of  its  exponents  in  the  terms  multiplied 
together; 

Determine  the  sign  of  the  result  by  the  rule  that  like  signs 
give  +,  and  unlike  signs  give  — . 

Ex.  1.    Multiply  5a^b3^  by  -  6a¥y\ 

The  product  is  -  30  a^¥x^y'^. 

Ex.  2.    Multiply  5a"+3  by  2a"-^ 

Since  n  +  3  and  n  —  1,  added,  give  2n  +  2, 
the  product  is  10a2"+2. 


EXERCISE 

15 

1. 

2. 

3. 

4. 

5. 

6. 

Multiply 

-5 

-3a 

Zah 

30a:y 

4a: 

-5a: 

by 

4 

-2 

-5 

-  1 

-2x 

-3a: 

7. 

8. 

9. 

10. 

11. 

12. 

Multiply 

?>ax 

-^xf 

lax 

-  5a26 

6c2^    - 

-  2x^yz 

by 

—  4:ax 

-Ixf 

-Zay 

-4c^2 

-3c^2    _ 

-  8a:2/V 

I 


MULTIPLICATION  OF  MONOMIALS  61 


i^ 

13. 

14.            15. 

16. 

17. 

18. 

Multiply 

4x^ 

^aa^         .5x 

2.12/3 

2|x3 

ix^ 

by 

.2x3 

ia'a^       mx 

.052/2 

io: 

.5x 

19. 

20. 

21. 

22. 

23. 

Multiply 

2n-l 

2n-l 

2n-l 

a;"~^ 

a;»-2 

by 

22  '            23 
Exs.    19-21  when  n  = 

2 

4.     Also 

x^ 

0^3 

Verify 

Exs.  22 

and  23 

when  n  = 

4  and  x 

=  3. 

24. 

25. 

26. 

27. 

28. 

Multiply 

a^x^-^ 

a^x--' 

a2a:"-3 

-  a'^x--^ 

^2n 

by  0^3          _  (j^^4           _  ^3^                aa:"+^        a:3n 

29.  30.                         31.                             32. 

5(a  +  6)3  3(a-j-by  -  6(a  +  6) .  7(a  +  by-^ 

2{a  +  6)2  -  (a  +  6)  -  2(a  +  6)3  3(a  +  bf 


33.   Multiply  2 "-2  by  2  and  verify  the  result  when  n  =  4:, 
■  34.   Write  out  all  the  factors  of  7a\    Of  (7a)\ 

35.  (aby  is  how  many  times  as  large  as  ab'^  when  a  =  3 
and  6  =  2? 

36.  How  many  ic's  are  there  in  the  product  of  5aa:3  j^y 
6a2x^?    How  many  a's? 

37.  How  much  money  do  five  empty  pocket-books'contain? 
5X0=? 

38.  Find  the  value  of  7  times  0.    Of  5a  X  0.    Of  0  x'^x^. 
Of  3(x  +  2/)  X  0. 

If  a  =  4,  6  =  f ,  c  =  0,  .T  =  1,  and  y  =  9,  find  the  value  of 

39.  abc.         ,..         41    5cx2/2.  43.   4c2  +  a. 

40.  a^c.  42.   a2  +  Scy,  44.    (3c  +  a;)3. 


62  SCHOOL  ALGEBRA 

ac  +  y  2a  +  c{x  +  y) 

45.     -3-.  47.     1 . 

^^    5ac^  +  l  ^g    5(a;  -  1)  +  8 

a:  -f  y  *  *  2a 

49.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

50.  How  many  examples  in  Exercise  3  (p.  19)  can  you 
now  work  at  sight? 

Multiplication  of  a  Polynomial  by  a  Monomial 
52.  Utility  of  the  Distributive  Law  in  Multiplication; 
Rule.  In  arithmetic  we  have  become  familiar  with  the  fact 
that,  for  instance,  5X67  =  5(60  +  7)=  5X60  +  5X7;  and 
that  this  principle  enables  us  to  perform  all  multiplications 
by  committing  to  memory  only  the  products  up  to  9  X  9. 

Similarly,  in  algebra,  a(6  +  c)  =  ah  +^c.    This  is  called 
the  Distributive  Law  of  Multiplication.    By  use  of  this  law, 
all  multiplications  in  algebra  can  be  performed  as  a  multi- 
plication of  pairs  of  monomials. 
Hence,  to  multiply  any  polynomial  by  a  monomial. 
Multiply  each  term  of  the  multiplicand  by  the  multiplier, 
and  set  down  the  results  as  a  new  polynomial. 
Ex.    Multiply  2a^  -  5a^b  +  3a¥  by  -  ^ab\ 

2a3  -  5a^b  +  Sab^         =4 

-Sab^ =  -  12 

Product    -  6a*b^  +  I5a^b^  -  Qa'b*   =  -  48 
The  check  is  obtained  by  letting  a  =  1,  and  6=2. 

EXERCISE   16 
1.  2.  3.  4. 

Multiply  2a  -\- 3x         3x  -  2y      4x^y  -  xy^         7 ax  -  4by 

by       Sax  —  5xy  2xy  —  3abxy 


MULTIPLICATION  OF  A  POLYNOMIAL  63 

Multiply: 

5.  Sac^  —  3m^n  by  5an.  9.  Sx""*"^  +  7a:"  by  —  4x. 

6.  m  —  m^  —  3m^  by  —  1m?n.  10.  3a:"-^  +  Sa;""^  by  7?, 

7.  8x^2/  —  5a:?/2  ^  ^\^y  3^2/.  11.  3a:^«  +  ^7?"  by  a:*". 
2a:"  -  3a:"~^  by  7?,  12.  2a^»  -  Ta^"  by  -  2a2». 


13. 

14. 

Multiply    2.5a:2  -  3.7a:  +  .51 

i^  -  ^^  -  i 

by           Ax 

i^ 

15. 

16. 

la,?  -  \ax  ~  |a 

Ax-^3?-l3?+  .5a;* 

-^ax 

-.25x' 

17.  What  is  the  value  of  7a:  —  by  times  zero? 
Multiply: 

18.  5(a  +  hf  -  3(a  +  6)  -  5  by  2(a  +  6). 

19.  7(a:  -  yf  +  2(a:  -  i/)  -  6  by  3(a:  -  y)\ 

20.  2(3a  +  26)2  _  5(3^  _|.  26)  +  4  by  4(3a  +  26). 

21.  Reduce  {7aaabb  —  5aaabb)  X  6aa66  to  its  simplest 
form.  Compare  the  size  of  the  result  with  that  of  the  original 
expression. 

(The  following  problems  are  mixed  variations  of  Types  I, 
n,  and  III.) 

22.  What  number  diminished  by  19  equals  37? 

23.  What  number  increased  by  19  equals  37? 

24.  What  number  diminished  by  1.067  equals  4.5? 

25.  What  number  increased  by  twice  itself  and  then  by 
24  equals  144?      ' 


64  SCHOOL  ALGEBRA 

26.  What  number  increased  by  twice  itself  and  then 
diminished  by  24  equals  144? 

27.  What  number  increased  by  J  of  itself  and  then  by  20 
equals  60? 

28.  What  number  diminished  by  J  of  itself  and  then 
increased  by  30  equals  90? 

29.  What  number  of  dollars  diminished  by  i  of  itself  and 
then  by  $30  equals  $160.60? 

30.  If  a  number  is  multiplied  by  3  and  then  diminished 
by  40,  the  result  is  140.    Find  the  number. 

31.  If  5  times  a  certain  number  is  increased  by  20.5,  the 
result  is  870.    Find  the  number. 

32.  If  five  times  a  certain  number  is  increased  by  20.5,  the 
result  is  equal  to  three  times  the  number  increased  by  160. 
Find  the  number. 

33.  A  man  who  died  left  $16,000  to  his  son  and  daughter. 
The  share  of  his  daughter,  who  had  taken  care  of  him  in  his 
illness,  was  $500  less  than  twice  the  share  of  the  son. 
How  much  did  each  receive? 

34.  A  cubic  foot  of  iron  and  a  cubic  foot  of  aluminum 
together  weigh  618  lb.  If  the  weight  of  the  iron  is  14  lb.  less 
than  three  times  the  weight  of  the  aluminum,  find  the  weight 
of  each. 

35.  A  baseball  nine  has  played  54  games,  and  the  number 
of  games  it  has  won  is  3  less  than  twice  the  number  it  has 
lost.    How  many  has  it  lost? 

36.  Of  which  type  is  each  of  the  above  problems  (Exs. 
22-35)  an  instance  or  a  variation? 


I 


MULTIPLICATION  OF  A  POLYNOMIAL  65 


Multiply  each  member  of  the  following  equalities  by  —  1 
and  solve: 


i^ 


37.    —  2x  —  5=— x-\-4:.        38.    —  4a:  —  a;  =  —  6  —  9. 


9.   Make  up  and  work  an  example  similar  to  Ex.  10.    To 
Ex.  19. 

40.  Make  up  and  work  an  example  similar  to  Ex.  30.  To 
Ex.  35. 

41.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

Multiplication  of  a  Polynomial  by  a  Polynomial 

53.  Arranging  the  Terms  of  a  Polynomial.  The  multi- 
plication of  polynomials  is  greatly  facilitated  by  arranging 
the  terms  in  each  polynomial  according  to  the  powers  of  some 
letter,  in  either  the  ascending  or  descending  order. 

Thus,  5x^  +S  —  X  +  X*  -  7x^,  arranged  according  to  the  as- 
ccDding  powers  of  x,  becomes 

S  -X  +5x'^  -7x^  +  x\ 

Also,  a'*  +  6'*  —  \.a^\P-  —  5a^b,  arranged  according  to  the  descend- 
ing powers  of  a,  becomes 

a*  -  5a'b  -  ^a%^  +  ¥. 

54.  Multiplication  of  Polynomials.  By  a  double  use  of 
the  Distributive  Law: 

(a  +  b)(c  +  d)  =  a(c  -i- d)  +  b{c  +  d) 
=  ac  -\-  ad  -{-  be  -\-  bd 

We  see  that  a  similar  result  is  obtained,  no  matter  how 
many  terms  occur  in  each  polynomial. 

Therefore,  to  multiply  two  polynomials,  ,u 


66  SCHOOL  ALGEBRA 

Arrange  the  terms  of  the  multiplier  and  the  multiplicand 
according  to  the  ascending  or  descending  powers  of  the  same 
letter; 

Multiply  each  term  of  the  multiplicand  by  each  term  of  the 
multiplier; 

Add  the  partial  products  thus  obtained. 

Ex.  1.    Multiply  2x  -  3y  by  3a:  +  5y. 

The  terms  as  given  are  arranged  in  order. 

The  most  convenient  way  of  adding  partial  products  is  to  set 
own  similar  terms  in  columns,  thus: 

2x   -    Sy  =  -  1 

Sx   +    5y =       8 

1 6x^  Qxy 

Partial  products  J        _^  ^^^^  _  ^^^^ 


Product  6^2  +      xy  -  15y^   =  -  8 

The  check  is  obtained  by  letting  x  =  1  and  y  =  l(ora:  =  y  =  1). 
Note  that  this  method  checks  only  the  signs  and  coefficients,  not 
the  letters  or  their  exponents.  Mistakes  in  letters  and  exponents, 
however,  are  rare  in  comparison  with  mistakes  in  signs  and  coeffi- 
cients. A  convenient  check  for  all  elements  in  the  process  is  ob- 
tained by  letting  x  =  y  =  2.  A  useful  check  on  the  letters  and 
exponents  in  many  examples  is  given  in  Art.  56. 

Ex.  2.    Multiply  2a:  -  o:^  +  1  -  Sa:^  by  2a;  +  3  -  x\ 

Arrange  the  terms  in  both  polynomials  according  to  the  ascending 
powers  of  x.  (Why  is  the  ascending  order  chosen  rather  than  the 
descending?) 


1  +2x  -Sx^  - 
S  -\-2x  -    x^ 

x^ 

-1 
4 

S  -\-6x  -9x^  - 

+  2a;  +  4a;2  - 

-    a;2  - 

Sx^ 

6x3  _  2x^ 

2x'  +  Sx'  +  x^ 

Product  3  +  8a;  -  6a;2  -  lla;^  +    x*  -^  x^   =  -  4 

Now  multiply  the  two  polynomials  together  with  their  terms  in 
the  order  as  first  given.  This  will  show  you  the  advantage  of  ar- 
ranging the  terms  in  order  before  multiplying. 


MULTIPLICATION  OF  A  POLYNOMIAL  67 

Ex.  3.  Multiply  a^-\-b^-\-c^  +  2ab-ac-bchy  a-\-b-\- c. 
Arranging  the  terms  according  to  powers  of  a, 


a' 
a 

+  2ab 
+     b 

+ 

ac 
c 

+ 

62 

-      bc-\-    & 

=  3 
=  3 

a' 

+  2a26 
+    a'b 

+ 

aH 

G?C 

+    a62 
+  2a62 

-  abc  +  ac^ 

-  abc 

-h  2abc  -  ac^ 

+  ¥ 

-¥c 
+  b^c 

+  6c2 
-bc^  +(^ 

a'  +  Sa^b  +  3a62  _j_  53  +  c^  =  9 

55.  Degree  of  a  Term;  Homogeneous  Expressions.  The 
degree  of  a  term  is  determined  by  the  number  of  literal  fac- 
tors which  the  term  contains.  Hence,  the  degree  of  a  term  is 
equal  to  the  sum  of  the  exponents  of  the  literal  factors  in  the 
term. 

Thus,  7a^bc'^  is  a  term  of  the  6th  degree,  since  the  sum  of  the 
exponents  in  it  is  3  +  1  +  2,  or  6. 

The  degree  of  an  algebraic  expression  is  the  same  as  the 
degree  of  that  term  in  the  expression  which  has  the  highest 
degree. 

Thus,  7x^  +  Sx^y"^  +  ?/  is  of  the  4th  degree. 

A  homogeneous  polynomial  is  a  polynomial  of  which  all  the 
terms  are  of  the  same  degree. 

Thus,  5a^b  —  ¥  +  ab^  is  a  homogeneous  polynomial,  since  each 
of  its  terms  is  of  the  3d  degree. 

56.  Multiplication  of  Homogeneous  Polynomials.  If  two 
monomials  are  multiplied  together,  the  degree  of  the  product 
must  equal  the  sum  of  the  degrees  of  the  multiplier  and  the 
multiplicand. 

For  instance,  in  Ex.  3,  above,  the  multiplicand  is  of  the  2d 
degree  and  the  multiplier  is  of  the  1st  degree,  and  are  both  homo- 
geneous. Their  product  is  seen  to  be  homogeneous  and  of  the  3(X 
4egree. 


68  DURELL'S  ALGEBRA:  BOOK  ONE 

The  fact  that  the  product  of  two  homogeneous  expressions 
mn^t  also  he  homogeneous  affords  a  partial  test  of  the  accuracy 
of  the  work. 

If,  for  instance,  in  Ex.  3,  p.  67,  a  term  of  the  5th  degree,  such  as 
5a%^,  had  been  obtained  in  the  product,  it  would  have  been  at  once 
evident  that  a  mistake  had  been  made  in  the  work. 

57.  Detached  Coefficients;  Symmetrical  Expressions.  The 
process  of  multiplying  algebraic  expressions  may  often  be 
further  abbreviated  by  using  only  the  signs  and  coefficients 
of  terms,  omitting  the  letters  and  their  exponents  and  thus 
saving  much  of  the  work  of  ordinary  multiplication. 

EXERCISE   17 

Multiply  and  check  each  result: 

1.  a:  -  4  by  2a;  +  1.  5.   7x^  -  5y^  by  Ax^  +  B?/^ 

2.  a;  -  3  by  3a:  +  2.  6.   5xy  +  6  by  Qxy  -  7. 

3.  2a;  +  5  by  a;  -  7.  7.   ^a^  -  b^c  by  Sa^c  +  2a6V. 

4.  3a;  -  42/  by  4a;  -  3y.      8.   ll^y  -  7xf  by  Sa;^  +  2y\ 
9.  a^  —  ab  -{-  ¥hy  a  +  b. 

10.  ^  -{-x^y  -[■  xy'^  -\-  fhy  x  -  y. 

11.  4a;3  _  3^2  _^  2a;  -  1  by  2a;  +  1. 

12.  2a;2  -  3xy  +  2/  by  3a;  -  5y. 

13.  a;^  -  3a;2  +  2a;  -  1  by  2ar^  +  a;  -  3/ 

14.  Sx^y  -  4a;2/2  -  i/^  by  a;^  -  2xy  -  y\ 

15.  a;^  —  ^x^y  +  Sxy^  —  fhy  x^  —  2xy  +  y\ 

16.  4a;3  _  33^  ^  5^  _  2  by  a;2  +  3a;  -  3. 

17.  a;^  -  3x2  +  5  by  a;2  -  a;  -  4. 

18.  3?  -dxy  -\-  fhy  a^  -  Sxy  -  f. 


I  MULTIPLICATION  OF  A  POLYNOMIAL  69 

9.  a"  -  ab +  ¥  by  0"  + ah +  b\ 

0.  4a:2  +  91/2  -  Qxy  by  ^x^  +  V  +  6a:y. 

21.  x'  -  rx'y^  +  6a;!/3  _  ^/^  by  a:^  _  2x1/2  _|_  ^^^ 

22.  3^  -  6aa:2  +  12a2a:  -  8a^  by  -  a:^  -  4aa:  -  Aa\ 

23.  a2  +  62  -j-  ic2  +  2a6  -  ax  -  6x  by  a  +  &  +  a;. 

24.  ab  +  cd  -\-  ac  -\-  bd  by  a6  +  cc?  —  ac  -  6c?. 
|H|5.  ia  +  46  by  Ja  -  |6. 

26.  !a:2-4a:  +  ibyfa:  +  |. 

27.  .5a  -  .46  by  .2a  -  .36. 

28.  1.8a:2  -  3.2a;  +  .48  by  2.5a:  +  .5. 

29.  x^  +  2x»-i  +  3a;"-2  -  2  by  a;  -  2. 

30.  a;"+^  -  3a;"  +  4a;"-^  -  Sa;"-^  by  a;"  +  2a;»-^ 

31.  a;"-^  -  2a;"-^  +  3a;"-2  -  4a;"-^  +  5a;"  by  2a,'2  +  3a;  +  1. 

32.  Multiply  3a;  -  5  +  4a;2  +  a;3  by  2a;  -  3  +  a;2  without 
changing  the  order  of  the  terms.  Now  arrange  the  terms  in 
each  expression  in  descending  order  and  multiply.  About 
how  much  easier  is  the  second  process  than  the  first? 

33.  Make  up  and  work  an  example  similar  to  Ex.  32. 
Arrange  the  terms  of  the  following  in  descending  order  of 

some  letter,  and  multiply: 

34.  4.T  -  3a;2  -  5  +  2a;3  by  a;  +  4. 

35.  3a;2  —  5  —  a;  by  a;  +  4a;2  —  2. 

36.  231^  -\-  y^  —  4iXy^  +  Sa^y  by  y^  +  Soi^  —  2xy. 

37.  Which  of  the  polynomials  in  Exs.  16-24  are  homo- 
geneous? 


70  SCHOOL  ALGEBRA 

38.  A  number  increased  by  3  times  itself  and  then  by  40 
equals  180.    Find  the  number. 

39.  Separate  180  into  two  parts  such  that  one  part  exceeds 
three  times  the  other  by  40. 

SuG.     Let  X  =  the  second  part. 

40.  A  number  increased  by  |  of  itself  and  then  by  20 
equals  95.    Find  the  number. 

41.  Separate  95  into  two  parts  such  that  one  part  exceeds 
§  the  other  part  by  20. 

42.  A  number  increased  by  f  of  itself  and  then  diminished 
by  30  equals  70.    Find  the  number. 

43.  Separate  70  into  two  parts  such  that  one  part  exceeds 
f  of  the  other  part  by  30. 

44.  A  number  din:  inished  by  |  of  itself  and  then  increased 
by  30  equals  66.    Find  the  number. 

45.  A  number  increased  by  .06  of  itself  and  then  by  $100 
equals  $312.    Find  the  number. 

46.  Separate  400  into  two  parts  such  that  one  part  exceeds 
3  times  the  other  part  by  60. 

47.  Separate  $1000  into  two  parts  such  that  one  part  is 
smaller  than  4  times  the  other  part  by  $100. 

48.  Of  which  type  is  each  of  the  above  problems  (Exs. 
38-47)  an  instance  or  a  variation? 

58.  Multiplication  Indicated  by  the  Parenthesis;  Simpli- 
fications. The  parenthesis  is  useful  in  indicating  multipK- 
cations  or  combinations  of  multiplications. 

Thus,  (a  -  b  +  2c)2  means  that  a  -  6  +  2c  is  to  be  multiplied 
by  itself. 

(a  —  b  +  2c)'  means  that  a  —  6  +  2c  is  to  be  taken  as  a  factor 
three  times  and  multiplied. 


1 


MULTIPLICATION  OF  A  POLYNOMIAL  il 


o  perform  the  multiplication  expressed  by  a  power  is  to 
expand  the  power. 

IBpAgain,  (a  —  b)  {a  —  26)  {a  -\-h  —  c)  means  that  the  three  factors, 
a  —  b,  a  —  2b,  and  a  -\-  b  —  c,  are  all  to  be  multiplied  together. 

Also,  (a  —  2xY  —  (a  +  2x)  (a  —  2x)  means  that  a  +  2x  is  to  be 
multiplied  by  a  —  2x,  and  the  product  is  to  be  subtracted  from  the 
m-oduct  of  a  —  2a;  by  itself. 

BPWe  simplify  an  expression  in  which  multiplications   are 

indicated  by  parentheses  and  exponents  by  performing  the 

operations  indicated  and  collecting  terms. 

Simplify  3(0-  -  2y)  {x  +  2y)  -  2{x  -  2y)\ 


\ 


Z{x  -  2y)  {x  +  2y)  -  2{x  -  2yY 

=  3(a;2  -  42/2)  -  2{x'^  -  4xy  +  4y^) 
=  3x2  _  12^2  _  (2a:2  -  8xy  +  Sy^) 
=  3^2  -  12?/2  -  2a;2  +  8xy  -  8y^ 
=  a;2  +  8xy  -  20y^  Ans. 


Check  this  result  by  letting  x  =  1  and  y  =  2. 

EXERCISE    18 

Find  the  product  of 

1.  (-  a)  (-  a)  (-  «)  (-  «)  (-  G^). 

2.  (-  1)  (-  1)  (-  1)  (-  1)  (-  1)  (-  1). 

3.  {x  -  y)  {x  -  y)  {x  -\-  y)  {x  -  y)  {x  +  y)  in  parenthe- 
sis form. 

4.  Find  the  value  of  (-  2)\    Of  a"  when  a  =  -  1  and 

71  =  7. 

Simplify  by  removing  parentheses  and  collecting  terms: 
t-5.   x-2{x-^  1).  Ue.    (2.r  +  3)  {^x  -  4). 

6.    {x  -  2)  {x  +  1).  9.  la  -  3(4a  -  8). 

^  7.   2t  +  3(5a:  -  4).  lo.   9a  +  5(3a  +  4). 


72  SCHOOL  ALGEBRA 

11.  3z(x  -  2)  -  2a:(a;  -  3). 

^  12.  (2a:2  -  3.T  +  1)^ 

A  13.  (2a  -  36  +  5)2  -  (2a  +  36  -  5)2. 

14.  {x  -  5)2  -  (a;  +  5)2. 

15.  3a:  -  2(3a:2  -  5x  +  2). 

16.  X  -  2(a:  -  1)  {x  +  3). 

17.  (a:  -2){x-  1)  (a;  +  3). 
^18.  3a2  -  (a  -  26)  (3a  +  46). 

19.  {x  —  y  —  zY  -  x(x  —  2y  +  2z). 

20.  2^2  -  3(x  -  1)2  +  {x  -  2)2. 

21.  3^2  -  x{l  -  x)(2  +  x)  +  a^. 

22.  2  -  3(.r  -  2)2  -  2(3  -  2x)  (1  +  x), 

23.  a2  —  [x{a  —  x)  —  a{x  ~  a)]  —  3?. 

24.  {x  -\){x-2)-{x-  2)  (a;  -  3)  +  (a:  -  3)  (a;  -  4). 

25.  3(a:  -  yf  -  2\{x -\-  yY  -  (x  -  y)  {x  +  y)}  +  2f, 

26.  x(x  -  y  -  z)  -  y{z  -  X  -  y)  -  z{z  -  y  -  x)  -  if. 

27.  3[(a  +  26)x  +  2my]  -  5[(m  -  c)y  +  6a:]  -  4[(a:  -  a) 
a  +  c?/]. 

28.  26a6  -  (9a  -  86)  (5a  +  26)  -  (46  -  3a)  (15a  +  46). 

29.  Multiply   the   sum   of    (a  —  2a:)2   and    (2a  —  a;)2  by 
3a  -  2(a  -  x), 

30.  Subtract  (x  —  2yy  from  a:^  —  8?/^  and  divide  the  re- 
mainder by  X  —  2y. 

31.  Find    the    value    of    3X0  +  4.      Of    8-7X0. 
Of  6  X  0  X  5  +  7. 


I 


MULTIPLICATION  OF  A  POLYNOMIAL  73 

If  a  =  3,  6  =  0,  X  =  —  2,  and  ?/  =  —  5,  find  the  values  of 

32.  2ax,  36.   by^  +  3x{x  -  y), 

33.  6a:^2/.  37.   4a;2  —  a6a:(4x  —  2/). 

34.  Sx^  +  a6i/.  38.   3a:  -  3(2x  +  3). 

35.  bxy  —  ax^.  39.   2{x^  +  y)  —  ciby  +  aic^. 

40.  2(1  -  2xy  +  (a:  +  2/)  (c^'  +  ^). 

41.  (a:  -  1)2  -  3(0:  +  1)  (a:  +  2)  -  xix"  -  2)  (y  -  2x). 

42.  Sa(a  -  2x)  -  {a  -  {a  -  1)  (x  +  1)  -  (a  +  x)^  +  5ax. 
Find  the  vahie  of 

43.  (x  -\-  ay  —  (x  —  ay  when  x  =  2a. 

44.  5(a;  +  py  —  (x  -\-  p)  {x  —  2p)  when  x  =  3p. 

45.  3a:2  _j_  4^  _  5(^;j.  _  ly  when  a:  =  ab. 

46.  If  a:  =  2  and  2/  =  1>  find  the  value  of  (x  +  yY.    Also 

of  a:^  +  ?/^. 

47.  From  the  sum  of  2a  +  56  and  36  —  5a,  subtract  three 
times  a  —  76,  and  verify  the  result  when  a  =  2  and  6  =  5. 
Also  when  a  =  3  and  6  =  —  1. 

48.  If  a  certain  number  is  diminished  by  24  and  the  result 
multiplied  by  3,  the  final  result  will  be  78.    Find  the  number. 

49.  If  a  certain  sum  of  money  is  increased  by  $150  and 
the  result  multiplied  by  4,  the  final  result  will  be  $1000. 
What  is  the  original  sum  of  money? 

50.  Separate  $1000  into  two  parts  such  that  one  part 
equals  four  times  the  sum  of  $150  and  the  other  part. 

51.  Separate  ..001 5  into  two  parts  such  that  one  part  equals 
3  times  the  sum  of  .0001  and  the  other  part. 


74  SCHOOL  ALGEBRA 

52.  The  sum  of  two  fractions  is  If,  and  the  larger  is  three 
times  the  sum  of  the  smaller  and  ^.    Find  the  fractions. 

53.  Separate  SI 00  into  two  parts  such  that  the  sum  of  one 
part  and  $10  equals  the  other  part. 

54.  Separate  $100  into  three  parts  such  that  3  times  the 
sum  of  $5  and  one  of  the  parts  equals  each  of  the  other  parts. 

55.  Separate  $100  into  four  parts  such  that  twice  the  sum 
of  one  part  and  $1  equals  each  of  the  other  parts. 

56.  A  man  walked  15  miles,  rode  a  certain  distance,  and 
then  took  a  boat  for  twice  as  far  as  he  had  previously  trav- 
eled. Altogether  he  went  120  miles.  How  far  did  he  go  by 
boat? 

57.  The  sum  of  three  numbers  is  50.  The  first  number  is 
twice  the  second,  and  the  third  is  16  less  than  three  times  the 
second.    Find  the  numbers. 

58.  Find  five  consecutive  numbers  whose  sum  is  3  less  than 
6  times  the  least  of  the  numbers. 

59.  The  difference  between  two  numbers  is  6,  and  if  3  is 
added  to  the  larger,  the  sum  will  be  double  the  less.  Find  the 
numbers. 

60.  Divide  $4500  among  two  sons  and  a  daughter  so  that 
each  son  gets  $100  less  than  twice  the  daughter's  share. 

61.  Find  two  numbers,  whose  difference  is  14,  such  that 
the  greater  exceeds  twice  the  less  by  3. 

62.  The  difference  of  the  squares  of  two  consecutive  num- 
bers is  43.    Find  the  numbers. 

63.  Three  boys  together  earned  $98.  If  the  second  earned 
$11  more  than  the  first,  and  the  third  $28  less  than  the  other 
two  together,  how  much  did  each  earn? 


MULTIPLICATION  OF  A  POLYNOMIAL 


75 


64.  Which  of  Exs.  48-63  are  instances  or  variations  of 
Type  I?    Of  II?    III? 

65.  Make  up  and  work  an  example  similar  to  Ex.  48.    To 
Ex.49. 

66.  Make  up  and  work  an  example  similar  to  Ex.  58.    To 
Ex.  62. 

67.  How  many  examples  in  Exercise  6  (p.  29)  can  you 
m  work  at  sight? 


.  CHAPTER  V 
DIVISION 

59.  Division  is  the  process  of  finding  one  factor  when  the 
product  and  the  other  factor  are  given. 

The  dividend  is  the  product  of  the  two  factors,  and  hence 
it  is  the  quantity  to  be  divided  by  the  given  factor. 
The  divisor  is  the  given  factor. 
The  quotient  is  the  required  factor. 

Thus,  to  divide  lOxy  by  5x,  we  must  find  a  quantity  which, 
multiplied  by  5a:,  will  produce  lOxy.  The  factor  5x  is  the  divisor, 
lOxy  is  the  dividend,  and  the  other  factor,  or  required  quotient,  is 
evidently  2y. 

The  division  of  a  by  6  may  be  indicated  in  each  of  the 
following  ways: 

h)a,    a  -^  b,    7,     or  a/b 

60.  General  Principle.  Division  being  the  inverse  of  mul- 
tiplication, the  methods  of  division  are  obtained  by  inverting 
thie  processes  used  in  multiplication. 

Division  of  Monomials 

61.  Index  Law  for  Division.  If  a^  is  to  be  divided  by  a^, 
we  have 

a^      aX  a  X  aX-GrX-et- 

Or,  in  general,  —  =  a"*-", 

(Ji 

where  m  and  n  are  positive  whole  numbers. 

76 


DIVISION  OF  MONOMIALS  77 

62.  The  Law  of  Signs  in  Division  is  obtained  by  inverting 
the  processes  of  multiplication. 

Thus,  in  multiplication,  if  a  and  h  stand  for  any  positive 
quantities  (see  Art.  50,  p.  59), 


■\-  aX  -\-h  =  -{-  ah] 
-i-aX— b=—  ab 
^aX  +b  =  -  ab 
-  a  X  ~  b  =  -{-  ab^ 


^+ab  ^  -\-b  =  -}-a...(l) 

-  ab  -^  -b  =  +  a...{2) 

-  ab  -ir  +b  =  -  a...{S) 
^-a6-^  -  6  =  -  a...(4) 


Hence,  by 
definition 
of    divi- 
sion. 

From  (1)  and  (2)  we  see  that  the  division  of  like  signs 
gives  +.  From  (3)  and  (4)  we  see  that  the  division  of  unlike 
signs  gives  —.  Hence,  the  law  of  signs  is  the  same  in  divi- 
sion as  in  multiplication. 

63.  Division  of  Monomials.  Combining  the  results  ob- 
tained in  Arts.  60,  61,  and  62,  we  have  the  following  method 
for  the  division  of  one  monomial  by  another: 

Divide  the  coefficient  of  the  dividend  by  the  coefficient  of  the 
divisor; 

Obtain  the  exponent  of  each  literal  factor  in  the  quotient  by 
subtracting  the  exponent  of  each  letter  in  the  divisor  from  the 
exponent  of  the  same  letter  in  the  dividend; 

Determine  the  sign  of  the  result  by  the  rule  that  like  signs  give 
plus,  and  unlike  signs  give  minus. 

Ex.  1.    Divide  27a36V  by  -  9a%7?, 

■ =  —  3a63  Quotient 

since  the  factor  x^  in  the  divigor  cancels  x^  in  the  dividend. 
Ex.2.    Divide  a2"-3  by-a--i. 

-~zr  =  a"*"^  Quotient 

Check  the  work  In  each  of  the  above  examples  by  multiplying  the 
quotient  bv  the  divisor. 


7S 


SCHOOL  ALGEBRA 


EXERCISE    19 

Di\dde  and  check  the  result: 

1.  15a  by  —  5a.  10.    —  m^n  by  —  m'. 

2.  -  3^:^  by  a-.  2.1.    -  3x2  by  _  1. 

3.  Sa^x^  by  —  4aa-2.  12.    —  8aar  by  Jar. 

4.  -  30a:^i/2  by  _  Gor^i/.    4*3.    166/  by  -  f  6y. 

5.  -  7x:^  by  72:^.  14.  8ma:  by  .2x, 

6.  2Lt?/22  by  —  3xz.  15.   .4aa:2  by  .8ar^. 
4.-J7.   186c^(f  by  —  9c^d.          16.   .04aa;  by  .Sox. 
UQ    -  33.TY27  by  lla-2/3^.     17.   2^3^  by  f.x-l 

9.  2Sx^y^^  by  —  Ma:?/^^^.     is.    —  jx^  by  .5a;. 

19.  47rr2by27r.    By  r^.    By  vrr. 

20.  ig^hygt    By  ig.    By  i<. 

21.  JmiJ^  by  fm.    By  .5v^,    By  .252?. 

22.  20(x  +  7/)3  by  -  4(a:  +  y).    By  -  2(a:  +  2/)'- 

23.  -  1.4(a  -  by  by  -  7(a  -  hf.    By  -  2(a  -  by. 
^  24.   a^''  by  a^".    B?/  a^".     —  a". 


25.    -  6a"+3by2a 


n+l 


Bva 


t+2 


-  3a 


n-l 


,2n-3 


26.  a2"+5  by  a"+^    By  a 


2n+3 


27.  How  many  2's  are  multiplied  together  in  2^''?    In  2*1 
In  the  quotient  of  2io  -^  2^? 

28.  How  many  a:'s  in  ic^°?     In  o:^?     In  the  quotient 

a:io  ~  a:^ 

29.  Divide  2""^  by  2  and  verify  your  result  when  n  =  5J 
Treat  2'^~^  -^  2^  in  the  same  way. 


DIVISION  OF  A  POLYNOMIAL  79 

30.  If  an  empty  box  is  divided  by  partitions  into  5  equal 
parts,  will  each  compartment  of  the  box  be  empty? 

31.  What  is  the  value  of  0  ^  5?  Of  0  ^  7?  State  the 
meaning  of  the  latter  in  a  manner  similar  to  that  used  in 
Ex.  30. 

32.  Give  the  value  of  0  -^  10.    Of  0  -^  a.    Of  0  ^  2x, 
0.0         0  0 


Of 


7a    lah    7ahx    la'h^T? 


3aa: 

33.  What  is  the  value  of  -—  when  a  =  0?    When  a;  =  0? 

72/ 

If  a  =  2,  6  =  3,  c  =  0,  a:  =  1,  find  the  value  of  each  of  the 
following: 

34.  *-«  36.    ^^?^^) 
a  4a 

35.  '^^  +  ^)  37.      ^^' 


h  '  h  +  x 

38.  What  is  a  polynomial?    A  binomial?    A  monomial? 
Give  two  examples  of  each.    . 

Division  of  a  Polynomial  by  a  Monomial 

64.  Utility  in  the  Distributive  Law  of  Division;  Rule.  In 

arithmetic  we  have  become  familiar  with  the  fact  that,  for 

instance, 

65  _  50  +  15  _  50  ,   15  _  .^  ,  ^  _  ,. 
-  -  _  +  _  _  10  +  3  -  Id; 

and  that  this  principle  enables  us  to  perform  all  divisions  by 
committing  to  memory  only  the  quotients  up  to  81  -^  9. 

Similarly,  in  algebra,  divisions  can  be  greatly  simplified  by 

the  fact  that 

axf-\-  he      dc  ,   he  ,   , 

= 1 =  a  -j-  b 

c  c        c 


80  SCHOOL  ALGEBRA 

This  is  called  the  Distributive  Law  of  Division. 
Hence,  to  divide  a  polynomial  by  a  monomial, 

Divide  each  term  of  the  dividend  by  each  term  of  the  divisor, 
I    and  connect  the  results  by  the  proper  signs. 

Ex.  1.  Divide  12a^x  -  lOa^y  +  6aV  by  2a\ 

i  2a^)12a^x  -  lOa^y  +  ea^z^ 
I  6ax  —       5y  +  Sa^z^    Quotient 

Ex.  2.  Divide  6a^^+^  -  4a2n+2  _  2a^n-s  by  2a^-\ 

3a2»+4  _  2a"+3_fl4n-2    Quotient 

Check  the  work  in  each  of  the  above  examples  by  multiplying  the 
quotient  by  the  divisor. 

I 

EXERCISE   20 
Divide  and  check: 

1.  a:^  —  Sa:^  by  —  x. 

2.  20^2  -  8xy  by  4a;. 

3.  4a62  -  Qa^bc  by  -  2ab. 

4.  -  Sa:^  +  7a;2  _  a:  by  -  x. 

5.  15x^y  —  lOx^y^  —  5xy^  by  5xy, 

6.  —  m  —  m^  -\-  m^  —  m^  by  —  m. 

7.  14::ji?yh  —  21a:i/V  +  xyz  by  —  xyz. 
a    -  3a:2  -  2a;  +  5  by  -  1. 
9.   .^3?  -  .12a:  +  9  by  -  .3. 

10.  .02a2  -  Mah  -  .Sb^  by  .5. 

11.  ix''  -  ix  -  I  by  -  |. 

12.  |a462  -  iaW  -  Wb'  by  -  f  a&. 


DIVISION  OF  A  POLYNOMIAL  81 

13.  9a:^"  -  6a;2»  +  12x«  by  -  3x\ 

14.  -   4a:2n+l  +   lOx^n+2    _    Q^n+2  ^y  2x^\ 

15.  a;»»+3  -  2a:»+2  _|-  3a;n+l  _|.  ^n  ^y  ^.n-l^ 

16.  8a:"»+2  _  i6a;"»+i  -  4a;"»  -  12a;^-^  by  -  4a:"»^^ 

17.  9x2«-2  -  6a:2»-i  ^  i2a:2"  -  3a:2"+i  by  3a;»-^ 

18.  10(a  +  6)2  -  8(a  +  6)  by  -  2(a  +  6). 

19.  .5(a;  -  yY  —  ,lb{x  —  yf  by  .5(a:  —  yY, 

20.  (a  +  6)a;  —  (a -f  6)?/  by  (a  +  6). 

21.  (a  —  h)x  -\-  {a  —  h)y  by  (a  —  6). 

22.  x(.T  +  1)  +  (a:  4-  1)  by  {x  +  1). 

23.  \  of  a  number  added  to  twice  the  number  gives  210. 

Find  the  number. 

24.  f  of  a  number  added  to  5  times  the  number  gives  340. 

Find  the  number. 

25.  f  of  a  number  added  to  J  the  number  gives  140.    Find 

the  number. 

26.  The  difference  between  f  and  §  of  a  certain  number  is 

14.    Find  the  number. 

27.  What  number  increased  by  .06  of  itself  gives  318? 

28.  What  sum  of  money  at  simple  interest  for  one  year  at 

6%  will  amount  to  $318? 

29.  What  number  increased  by  .15  of  itself  will  amount  to 

690? 

30.  What  sum  of  money  at  simple  interest  at  5%  will 

amount  to  $690  in  3  years? 

31.  For  every  nickel  which  a  girl  put  in  her  savings  bank 

her  father  put  in  a  dime.    If  her  bank  contained  $18.75  at 


82  SCHOOL  ALGEBRA 

the  end  of  one  year,  how  many  nickels  did  the  girl  save  in 
that  time? 

32.  For  every  dime  that  a  boy  spent  for  books,  his  father 
gave  him  a  quarter  to  spend  for  the  same  purpose.  If  he 
spent  $52.50  in  all,  how  much  did  his  father  give  him? 

33.  A  purse  contains  $10.50  in  dollar  bills  and  quarters, 
but  there  are  twice  as  many  quarters  as  bills.  How  many 
are  there  of  each? 

34.  How  can  $2.25  be  paid  in  5  and  10  cent  pieces  so  that 
the  same  number  of  each  is  used? 

35.  How  can  $5.95  be  paid  in  dimes  and  quarters  using 
the  same  number  of  each? 

In  the  following  equations  divide  each  member  by  —  1 
and  solve,  checking  each  result: 

36.^  -  1  -  3a:  =  -  a:  -  5       38.    -  x  -{2x  -  1)  =  -  5 

37.    -5x-8-x=-7x+l       39.    -  7a:  -  5  =  -  3a:  +  4 

40.  How  many  of  Exs.  23-35,  pp.  81-82,  belong  to  Type 
I?     To  Type  II?     III? 

41.  Make  up  and  work  an  example  similar  to  Ex.  31.  To 
Ex.  36. 

42.  Make  up  and  work  an  example  similar  to  Ex.  13.  To 
Ex.  18. 

43.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

Division  of  a  Polynomial  by  a  Polynomial 

65.  General  Method.  The  method  of  dividing  one  poly- 
nomial by  another  is  to  arrange  the  polynomials,  according 
to  the  ascending  or  descending  powers  of  some  one  letter. 


I 


DIVISION  OF  A  POLYNOMIAL  83 


and  then,  in    effect,  to    separate    the    dividend  into  par- 
tial   dividends,    which  are  divided   in   succession   by   the 
divisor. 
Ex.  1.    Divide  Qx^ -j- 7a^  -  3x^  +  Ux  -  6  by  2x^  +  3a:-2. 

We  divide  the  first  term  of  the  dividend,  6x*,  by  the  first  term 
of  the  divisor,  2x^,  obtaining  the  quotient  Sa:^.  Multiplying  this 
quotient,  Sx^,  by  the  entire  divisor,  we  obtain  the  first  partial  divi- 
dend. If  we  subtract  this  from  the  entire  dividend  and  divide  the 
remainder  by  2x^y  we  have  a  process  like  the  following: 

Dividend                      Divisor 
A , ^ , 


Qx^  +  7x^  -  Sx^  +  llx  -  Q\2x^  H-3a;  -2=  15  -v-  3 
Qx*  +  9x3  _  6a;2  3x2-    xj-3=    5 

-  2x3  ^  3a;2  4_  112;  _  6  ^~y~^ 
~2x3-3x^+    2x  «^^^^^ 

6x2  ^    Qx  -Q 
6x2  4,    93.  _  6 

A  quick  check  on  the  parts  of  the  work  in  which  errors  are  most 
likely  to  be  made  is  obtained  by  letting  x  =  1,  as  is  done  in  the 
solution  above.  A  more  complete  check  is  obtained  by  finding  the 
product  of  the  divisor  and  quotient  and  noting  whether  the  result 
equals  the  dividend. 

Now  state  the  process  of  dividing  a  polynomial  by  a 
polynomial  as  a  general  rule. 

Ex.  2.  Divide  Slah^  -  206^  -  lOa'b^  +  Qa^  -  a^b  by 
3a2  -  562  ^  4^5^ 

6a*  -    a%  -  10a%^  -f  Slab^  -  206^3^2  +  4a6  -562  =6^-2 
6a*  +  Sa%  -  10a262 2a^  -  Sab  +  462  =  3 

-  9a36  +  31a6' 

-  9o36  -  12^262  +  I5a63 


+  12a262  +  16a63  -  206* 
+  12a262  +  16a63  -  206* 


Is  the  dividend  homogeneous?    The  divisor?    The  quotient? 


84  SCHOOL  ALGEBRA  Jj 

Ex.  3.    Divide  x^  +  2/^  +  s^  +  Sx^y  +  Sxy^  hy  x  +  y  +  z. 

Arranging  terms  according  to  the  descending  powers  of  x, 
x'  +  Sxhj  +  Sxy^  +     2/3  +  z^x  +y  -{-z  =9-5-3 

^  +   x^y  +   xH x^  +  2xy  - xz  +  y^  +  z^  -yz  -•  3 

+  2xh/  -    x^z  +  Sxy^  +  2/'  +  2' 

-f  2x^  +  2xy^  +  2xyz 


-  x-'z  +   xy^ 

-  xH  -    xz^ 

-  2xyz  +  7/  +  2' 

-  xyz 

xy^  +  xz^ 
xt 

-    xyz  +2/^+2' 

+  y^+  y^z 

•Vxz^ 
-       -        +xz^ 

-  xyz            -y^z  +z^ 

-  xyz  -  yH  -  2/22 

-  xyz  -  y'^z  -  yz^ 

The  process  of  algebraic  division  may  often  be  abbreviated 
by  the  use  of  detached  coefficients.    (See  Appendix,  p.  466.) 

EXERCISE   21 
Divide  and  check  each  result: 

1.  3r^  +  7x  +  2bya;  +  2. 

2.  6a:2  +  7a:  +  2by3a;  +  2. 

3.  12x2  -i-xy  -  20?/2  by  3a:  -f  4?/. 

4.  3a:2  +  X  -  14  by  a;  -  2. 

5.  6a:2  _  31^2/  +  352/  by  2a;  -  7y, 

6.  12a2  -  1  lac  -  36c2  by  4a  -  9c. 

7.  -  15a:2  -f  59a;  -  56  by  3a;  -  7. 

8.  44a;2  —  xy  —  3y^hy  llx  —  Sy. 

9.  a2  -  462  by  a  -  26.       12.  9a;2  -  49  by  3a;  -f  7. 

10.  a^  —  y^hy  X  —  y.  13.   125  —  64a;^  by  5  —  4a;. 

11.  27a;3  +  8  by  3a;  +  2.       14.  8aV +  / by  2aa;  +  2/^. 


DIVISION  OF  A  POLYNOMIAL  85 

15.  2a:3  __  9^  ^  H^  _  3  by  2a:  -  3. 

16.  353^  +  47a:2  +  13a:  +  1  by  ox  +  1. 

17.  6a3  -  ITa^a:  +  Uax"^  -  ^^  by  2a  -  3a:. 

18.  iy"  -  W  +  222/2  -  72/  +  5  by  2i/  -  5. 

19.  (^  +  c*x  +  (^x'^  +  c^a:^  _^  ^^4  _^  ^5  ^y  ^  _|_  ^.^ 

20.  11a:  -  8a:2  _|_  5^  _  20  +  2a:4  by  a:  +  4. 

21.  4a:  +  6a:^  +  3a:2  -  lla:^  _  4  ^y  80^2  _  4. 

22.  -  a:^2/  -  lla:2/^  -  2a:y  +  Ga:*  -  ^y^  by  2a:  -  3y. 

23.  42/3  _^  6^6  _  13^4^^  by  3a:2  -  2z/. 

24.  a:^  -  16?/*  by  a:  -  2y.      26.  a:^  -  /  by  a:  +  t/- 

25.  a:^  +  32?/5  by  a:  +  2?/.       27.  256a:8  -  i/S  by  4a:2  _  ^^2^ 

28.  9a:  -  ISa:^  +  Sa:^  -  ISa:^  +  2  by  4a:2  +  x  -  2. 

29.  10  -  x^  -21x^  +  12a:4  -  3a:  by  a:  +  4a:2  -  2. 

30.  22a:2  _  13^  _|_  i^^  _  13^4  _|_  5^.  _  g  by  a:  +  Sa:^  -  2. 

31.  14a:V  -  16^:^2/^  +  6a:^  +  2/^  +  5a:42/  "  6ar2/^  by  3a:2  _^  ^/^ 
-2a:2/. 

32.  ba'h  -  '^aW  -  aW  +  3a^  -  4&^  by  a"  +  3a6  +  262. 

33.  a:^  —  2/^  +  2^  —  3:2/2  —  2a:22  +  2yz^  hy  x  —  y  —  z, 

34.  c^  +  ^3  +  n^  —  3ccZn  by  c  +  ^  +  n.  ,• 

35.  2/'  -2f  +  l  by  2/2  -  22/  +  1. 

36.  2x«  +  1  -  3x^  by  1  +  2a:  +  a:2. 

37.  Qxhf  -  62/2^2  -  6a:2s2  -  iSxyz^  -  bxyh  by  ^xy  +  2yz ; 
+  3a:z. 

38.  x^  -  39a:  +  15  -  2a:3  by  Zx^  j^  ^x -^  ;>? -\- 15. 

ao    4^6  -  ^a:*  +  25  -  14a:3  _  ^.2  by  2a:3  _  ^  _  5  _l_  3^^^ 


86  SCHOOL  ALGEBRA 

40.  Ja2  _  152  by  la  +  16. 

41.  ia;2  -  1^2/^  by  ^x  -  \y. 

42.  \a^  -  jj¥  by  ia  -  |6. 

43.  i:^  -  -V-^  +  f  by  ix  -  2. 

44.  A6x^  —  .25y^  by  Ax  +  .5y. 

45.  2.880:^  +  10.86a;  -  19.2  by  1.2^^  +  1.5x-j-  6.4. 

46.  6.t2«+i  -  13a:2n  ^  Q^r^"-!  by  3a:"+i  _  2x». 

47.  12.T^"  +  ISo:^"  -  a;"  by  3.T"  +  1. 

48.  4a;»+3  +  5a:»+2  _  ^n+i  _  a;n  ^  ^.n-i  j^y  3.2  _|_  2^  +  1. 

49.  6a-+i  -  5a:"  -  6a:"-i  ^  i3a.n-2  _  g^n-s  ^y  2.r2  -  3a:  +  2. 

50.  In  Ex.  20  try  to  divide  without  arranging  the  terms  of 
the  dividend  either  in  ascending  or  descending  order. 

51.  What  is  the  value  of  — ^  when  a:  =  0?    When  2/  =  0? 

x-\-  y 

52.  Divide  a:^  +  2/^  bya:  —  y  to  5  terms  and  note  the 
remainder. 

53.  Divide  1  by  1  —  a:  to  4  terms  and  note  the  remainder. 

54.  Divide  1  by  1  —  aa:  to  3  terms. 

55.  If  a  boy  walks  at  the  rate  of  3  miles  an  hour,  how  far 
will  he  walk  in  5  hours?  In  a  hours?  In  x  hours?  In  a:  +  2 
hours? 

56.  A  boy  starts  at  a  given  time  and  walks  5  hours.  An- 
other boy  then  starts  and  rides  a  bicycle  x  hours  until  he  over- 
takes the  first  boy.  How  many  hours  does  the  second  boy 
ride?  How  many  if  the  first  boy  has  a  start  of  a  hours?  Of 
y  hours? 


DIVISION  OF  A  POLYNOMIAL  87 

57.  Two  men  A  and  B  start  from  places  35  miles  apart 
and  walk  toward  each  other  at  the  rate  of  4  miles  and  3 
miles  an  hour  respectively.  How  many  hours  will  it  be 
before  they  meet? 

SuG.  In  forming  an  equation,  it  is  an  aid  to  diagram  a  problem 
of  this  kind :  35^1^ 

If  the  two  men  start     j^(^  ^  )^ 

at  the   same  time   and  ^^  | 

walk  toward  each  other 
•intil  they  meet,  they  must  travel  the  same  number  of  hours. 

Let  X  =  the  number  of  hours  each  man  travels. 

Then         4x  =  number  of  miles  A  travels. 
3a;  =  number  of  miles  B  travels. 
4x  -^Sx  =  35  (Art.  15,  1) 

7x  =  35 
X  =    5,  no.  hours  before  they  meet. 

Check.      4x  =  20,  distance  A  travels. 

3a;  =  15,  distance  B  travels.  ; 

20  +  15  =  35  ' 

In  working  Exs.  58-72,  draw  a  diagram  as  an  aid  in  each 
solution: 

58.  Two  men,  A  and  B,  start  from  places  42  miles  apart 
and  walk  toward  each  other,  at  the  rate  of  4  and  3  miles  per 
hour  respectively.  How  many  hours  will  it  be  before  they 
meet? 

59.  Make  up  and  work  an  example  similar  to  Ex.  58. 

60.  Two  bicyclists,  A  and  B,  start  respectively  from  New 
York  and  Philadelphia,  90  miles  apart,  and  ride  toward  each 
other.  A  rides  8,  and  B,  12  miles  per  hour.  How  long  and 
how  far  will  A  ride  before  meeting  B? 

61.  Boston  is  234  miles  from  New  York.  If  two  automo- 
biles start  from  the  two  cities  at  the  same  time  and  travel 


88  SCHOOL  ALGEBRA 

toward  each  other  at  the  rate  of  12  and  14  miles  per  hour 
respectively,  how  far  will  each  go  before  they  meet? 

62.  Make  up  and  work  a  similar  example  concerning  trains 
which  travel  between  New  York  and  Chicago,  which  are  912 
miles  apart. 

63.  One  boy  starts  at  a  certain  time  from  New  York  on 
a  bicycle  and  travels  toward  Philadelphia  at  the  rate  of  8 
miles  an  hour.  One  hour  later  another  boy  starts  from 
Philadelphia  and  goes  toward  New  York  at  the  rate  of  6 
miles  an  hour.     How  long  before  they  will  meet? 

64.  New  York  and  Washington  are  228  miles  apart.  A 
train  starts  from  New  York  at  a  given  time  and  goes  at  the 
rate  of  26  miles  an  hour,  and  two  hours  later  a  train  starts 
from  Washington  and  proceeds  at  the  rate  of  34  miles  an 
hour.    How  long  before  they  will  meet? 

65.  Make  up  and  work  an  example  similar  to  Ex.  64  con- 
cerning trains  which  travel  between  Cincinnati  and  New 
Orleans,  which  are  830  miles  apart, 

66.  Two  boys  start  at  the  same  place  and  travel  in  oppo- 
site directions  on  bicycles  at  the  rate  of  8  miles  and  10  miles 
an  hour.    How  long  before  they  will  be  108  miles  apart? 

67.  If  they  travel  in  the  same  direction,  how  long  before 
they  will  be  16  miles  apart? 

68.  Two  boys  start  from  New  York  and  Philadelphia  at 
the  same  time  and  travel  toward  each  other  until  they  meet. 
If  one  goes  twice  as  fast  as  the  other  and  they  meet  in  7^ 
hours,  what  is  the  rate  per  hour  of  each  boy? 

69.  If  in  Ex.  68  one  boy  went  5  miles  an  hour  faster  than 
the  other,  and  they  met  in  6  hours,  what  was  the  rate  of  each? 


REVIEW  89 

70.  Make  up  and  work  an  example  similar  to  Ex.  68 
concerning  automobiles  traveling  between  New  York  and 
Washington. 

71.  Make  up  and  work  an  example  similar  to  Ex.  69  con- 
cerning railroad  trains  traveling  between  New  York  and 
Buffalo,  which  are  440  mi.  apart. 

72.  A  set  out  from  a  town,  P,  to  walk  to  Q,  45  miles  distant, 
an  hour  before  B  started  from  Q  toward  P.  A  walked  at  the 
rate  of  4  miles  an  hour,  but  rested  2  hours  on  the  way;  B 
walked  at  the  rate  of  3  miles  an  hour.  How  many  miles  did 
each  travel  before  they  met? 

73.  How  many  examples  in  Exercise  10  (p.  45)  can  you 
now  work  at  sight? 

EXERCISE  22 

Review 

1.  Express  the  following  in  as  few  terms  as  possible:  3.2x2  — 
2.6xy  +  .16t/2  +  1.5x2  -  .8y^  -  .S2xy  +  Ay^  -  l.bx^  +  Axy. 

2.  Subtract  .ISa^  +  .362  _  2.5a6  from  -  7a?  -  4ab  -  1.5&2. 

3.  Add  2ip2  -  l.5p  +  I,  .75p2  +  |p  -'.4,  |  -  ^If-  -  .5p. 

4.  Simplify  3.2^2  -  [.8^2  +  (3.5x  -  J  -  .2x2)  -  1.5  -  3x]. 

5.  Solve  .3x  -  4  =  .2x  +  .5, 

6-   What  is  the  root  of  an  equation?    How  do  you  check  your 
solution  of  an  equation?    Check  Ex.  5. 

7.  Simplify  5x  -  3(x  -  2)  (x  +  7)  +  3(x  -  2)2. 
If  a  =  0,  6  =  1,  c  =  4,  X  =  -  2,  find  the  value  of 

8.  a{b  +c)  -  3x. 

9.  (c  +  2x)  (b  -a)  -3(x  +  4)  (x  +  5). 
^Q    3a  +  5(2  4-  x) 

b  +c 


90  SCHOOL  ALGEBRA 

11.  Multiply  a;  -  2  +  4a;2  by  2^2  -  1  -  5a;. 

12.  Divide  x*  +  S  -  Gx^  +  x^  +  8x  -  llx'  by  2x  -  x^  +  3. 

13.  Find  three  consecutive  numbers  whose  sum  is  33. 

14.  In  a  certain  kind  of  concrete,  twice  as  much  sand  is  used  as 
cement,  and  twice  as  much  gravel  as  sand.  How  many  pounds  of 
each  are  used  in  making  2800  lb.  of  concrete? 

15.  The  record  time  for  the  100  yd.  swim  at  a  certain  date  was 
55f  sec.  This  was  7§  sec.  more  than  5  times  that  for  the  100-yd. 
dash.    What  was  the  record  time  for  the  latter? 

16.  Solve  Ex.  15  without  using  x  to  represent  the  unknown 
number.  How  much  of  the  labor  of  writing  out  the  solution  is 
saved  by  the  use  of  x?  Is  there  any  other  advantage  in  using  x  to 
solve  problems? 

17.  What  is  the  dividend  when  the  quotient  is  a;^  +  2x^  +  7x 
+  20,  the  remainder  62a;  +  59,  and  the  divisor  a;^  —  2a;  —  3? 

18.  What  is  the  divisor  if  the  quotient  is  x^  +  3a;,  the  dividend 
x^  —  8,  and  the  remainder  9a;  —  8? 

19.  If  a;  =  —  J  and  y  =  —  i,  find  the  value  of 


(3a;  -  2yy  {9x^  +  ^y^)  -  6(y  -  x)  VGxy{x  +  2y^  +  4). 

20.  Add  a  to  h.    Also  add  3a  -  56  to  4c  +  7d. 

21.  Subtract  Sx  —  2y  +  z  from  —  7.    From  —  a.    From  h. 

22.  Subtract  2a  -  36  from  0.    Also  5  from  0. 

23.  Can  3  +  2a6  be  united  in  a  single  term?    Give  a  reason. 

24.  The  product  of  an  even  number  of  negative  factors  has  what 
sign?  Of  an  odd  number  of  negative  factors?  Give  an  example 
using  not  less  than  five  factors. 

25.  Express  the  following  in  a  simpler  form:  6aaa{x  —  y)  ix  —  y) 
{x  -y){x  -  y). 

26.  If  a  boy's  mark  on  each  of  three  recitations  is  0,  what  is  his 
average  on  the  recitations?   Give  the  value  of  0  +  0  +  0.    Of  3  X  0. 

0,| 

27.  Find  the  value  of  8  X  0  -  5  +  ^. 


REVIEW  91 

28.  Form  the  power  whose  base  is  5  and  exponent  2.  Also  the 
power  whose  base  is  2  and  exponent  5.  Find  the  difference  in  value 
between  these  two  powers. 

29.  Find  the  value  of  each  of  the  following  products  and  verify 
each  result  for  the  values  x  =  2,  a  =  S,  b  =  1. 

(1)  a:«.a;2«  (2)  x'^  +  K  x"-^  (S)x^  +  \x^-^ 

30.  From  the  product  of  Sx^  —  2  and  2x  —  5  subtract  7  times 
the  product  of  x  and  x  —  2. 

31.  Show  on  squared  paper  that  3x4+5x4=8x4. 
Also  that  4X5+7X5-2X5=9X5. 

32.  Multiply  1^2  -  ax  -h  fa^  by  fx^  +  ^ax  +  ia\ 

33.  Divide  x^"  -  2/^"  by  x""  -  ?/". 

34.  Divide  2  —  a;  by  1  +  re  to  five  terms  in  the  quotient. 

35.  Divide  [{x''  -  2x  -  1)  {x  -  1)  -{-  2x'  -  2x\  by  {{x  +  2) 
(a;  +  1)  -  {x^  +  2a^  +  3)]. 

36.  Multiply  Z.2x^  -  4..bxy  +  l.Si/^  by  1.5a;  -  3.52/. 

37.  Divide  a;^  —  15  by  a;^  +  a;  —  1  to  five  terms. 

38.  Divide  36^2  j^  W  +  \  -  ^^V  -  6a;  +  ly  by  6x  -  Jy  -  i 

39.  Divide  2.4a;3  -  {).l2x'^y  +  4.322/^  by  1.5a;  +  l.St/. 

40.  Divide  a?  +h^  -\-(?  -  Sabc  by  a  +  6  +  c. 
Solve  and  verify 

41.  (2a;  +  1)  (a;  -  3)  +  7  =  a;  -  2(a;  -  4)  (2  -x). 

42.  7a;  -  2(a;  -  1)  (2  -  a;)  -  17  =  a;(3a;  +  7)  -  (a;  +  l)'. 
Simplify: 

43.  62  +  [40  -  {8x  -  (22  +  4a;)  -  22a;!  -  7a;]  -  [7a;  +  [142  - 
(42  -5a;)}].       - 

44.  a'^ih  -c)  -  h\a  -  c)  +  c\a  -  b)  -  (a  -  b)  (a  -  c)  (b  -  c). 

45.  What  is  the  advantage  of  regarding  a  polynomial  as  made  up 
of  terms?    (What-  is  a  polynomial?    A  term?) 

46.  What  is  the  name  of  an  expression  containing  two  terms? 
Three  terms? 

47.  Write  a  homogeneous  expression  containing  three  terms,  and 
*Jie  letters  x  and  y. 


92  SCHOOL  ALGEBRA 

48.  If  s  =  ar"-*,  find  the  value  of  s  when  a  =  2,  r  =  3,  n  =  4. 
Also  when  a  =  2,  r  =  1,  and  n  =  5.   When  a  =  3,  r  =  J,  and  n  =  5. 

49.  Who  first  used  the  letters  x,  y,  and  z  to  represent  unknown 
numbers  in  equations  in  algebra?  (See  p.  455.)  Find  out  all  you 
can  about  this  man. 

50.  Give  some  of  the  other  symbols  that  were  used  to  represent 
numbers  before  the  use  of  the  three  last  letters  of  the  alphabet  was 
suggested. 

51.  Can  you  point  out  any  advantages  in  the  use  of  x,  y,  and  z 
instead  of  the  other  symbols  once  used  for  the  same  purpose? 

52.  Find  out,  if  you  can,  whether  any  other  symbols  than  the 
last  letters  of  the  alphabet  are  now  used  to  represent  an  unknown 
number  in  an  equation?  How  many  dififerent  symbols  can  be  used 
for  this  purpose? 

53.  Who  invented  the  parenthesis  sign  and  when? 

54.  How  many  examples  in  Exercise  13  (p.  54)  can  you  work  at 

sight? 


CHAPTER  VI 

EQUATIONS  (continued) 

66.  The  Equation,  members  of  an  equation,  and  transpo- 
sition have  already  been  explained.  (See  Arts.  42-45,  pp. 
52-53.) 

67.  A  Root  of  an  equation  is  a  number  which,  when  substi- 
tuted for  the  unknown  quantity,  satisfies  the  equation;  that 
is,  reduces  the  two  members  of  the  equation  to  the  same 
number. 

Ex.    If  in  the  equation,  3a;  —  1  =  2a:  +  3, 
we  substitute  4  in  the  place  of  x  in  each  member, 
we  obtain  3x  -  1  =  12  -  1  =  11 

2x +3=8+3  =  11 
The  equation  is  satisfied.   Hence,  4  is  the  root  of  the  given  equation. 

68.  The  Degree  of  an  Equation  having  One  Unknown 
Quantity.  If  an  equation  contains  only  one  unknown  quan- 
tity, the  degree  of  the  equation  (after  the  equation  has  been 
reduced  to  its  simplest  form)  is  determined  by  the  highest 
exponent  of  the  unknown  quantity  in  the  equation. 

Thus,  if  X  is  the  only  unknown, 

2a;  +  1  =  5a;  —  8  is  an  equation  of  the  first  degree. 
ax    =  b^   -\-  ex  is  of  the  first  degree. 

4x^  —  5x  =20         is  of  the  seeond  degree. 
3x2  _  3.3   =  6a;  +  8  is  of  the  third  degree. 

A  simple  equation  is  an  equation  of  the  first  degree. 

An  equation  of  the  first  degree  is  also  often  termed  a  linear 
equation,  for  reasons  which  will  be  explained  later.    (See  Art.  148.) 

93 


94  SCHOOL  ALGEBRA 

69.  Identities  and  Conditional  Equations.  If  we  take  the 
expression  (x  —  2)  (x  +  2)  =  x^  —  4,  and  substitute  x^=  1, 
we  obtain  —  3  =  —  3.  The  two  members  of  the  expression 
are  found  to  be  equal. 

Similarly,  they  are  found  to  be  equal  if  we  let  x  =  2,  3,  4, 
etc.;  0,-1,-2,  etc.;  or  any  number.  An  expression  hav- 
ing this  characteristic  is  termed  an  identity.  I 

An  identity  (or  identical  equation)  is  an  equality  whose 
two  members  are  equal  for  all  values  of  the  unknown  quantity 
(or  quantities)  contained  in  it. 

A  conditional  equation  is  an  equation  which  is  true  for 

only  one  value  (or  a  limited  number  of  values)  of  x.    For 

the  sake  of  brevity,  a  conditional  equation  is  usually  termed 

an  equation. 

The  equations  studied  in  Art.  42  (p.  52)  and  Exercise  13  (p.  54) 
are  conditional  equations. 

Hence,  the  sign  =  is  used  in  two  senses  in  elementary 
algebra,  viz.:  to  indicate  sometimes  an  equation,  and  some- 
times an  identity.  The  context  enables  us  to  decide  readily 
which  of  these  two  meanings  the  sign  =  has  in  any  given  case. 

Later  it  will  be  found  useful  to  use  the  mark  =  to  indicate  an 
identity,  and  =  to  indicate  a  conditional  equation,  or  equation 
proper. 

70.  The  Aids  in  Solving  an  Equation,  given  in  Art.  15, 
p.  18,  stated  more  precisely,  are  as  follows: 

The  roots  of  an  equation  are  not  changed  if 

1 .  The  same  quantity  is  added  to  both  members  of  the  equation. 

2.  The  same  quantity  is  subtracted  from  both  members  of  the 
equation. 

3.  Both  members  are  multiplied  by  the  same  quantity  or  equal 
quantities  (provided  the  multiplier  is  not  zero,  or  an  expression 
containing  the  unknown). 


EQUATIONS  95 

4.  Both  members  are  divided  by  the  same  quantity  (provided 
the  divisor  is  not  zero,  or  an  expression  containing  the 
unknown). 

Other  principles  similar  to  these  are  used  later  as  aids  in  solving 
equations. 

Transposition  (see  Art.  45,  p.  52)  is  a  short  way  of  using  Prin- 
ciples 1  and  2  of  this  article. 

71.  The  Method  of  Solving  a  Simple  Equation  may  now 
be  stated  as  follows: 

Clear  the  equation  of  parentheses  by  performing  the  operations 
indicated  by  them; 

Transpose  the  unknown  terms  to  the  left-hand  side  of  the 
equation,  the  known  terms  to  the  right-hand  side; 

Collect  terms; 

Divide  both  members  by  the  coefficient  of  the  unknown  quantity. 

'  Ex.    Solve  x{x  -  2)  =  x(a;  +  4)  -  3(a:  -  3) (1) 

Removing  parentheses,  x^—2x=  x^  +4a;  —Sx+9 

Transposing  terms  (Art.  70,  1,  2),  x'^  -x"^  -2x  -4tx-{-Zx  =9 
Collecting  terms,  —3a;  =9.  .  .  .(2) 

Dividing  by  -3  (Art.  70,  4),  a;=  -3  Root 

Check.    x{x  -2)  =  -  3(  -  3  -  2)  =  -  3(  -  5)  =15 
a:(x  +  4)  -  3(a;  -  3)  =  -  3(-  3  +  4)  -  3(-  3  -  3) 

=  -3  -3(-6)  =  -3  +  18  =  15 

EXERCISE  23 

Solve  the  following;  refer  to  each  principle  in  Art.  70  as 
you  use  it,  and  check  each  answer: 

1.  2a:  =  15  -  3a:.  6.  3.T  -  7  =  14  -  4a:. 

.    2.   15  +  3a:  =  27.              •  7.  2a:  -  7  =  8  +  5a:. 

3.  4a:  -  11  =  29.  8.  2a:  -  (a:  -  1)  =  5. 

4.  16a:  +  3  =  15a:  +  7.  9.  2  ft.  +  a:  =  12  ft. 

5.  14a:  -  10  =  12a:  -  3a:.  lo.  7  in.  +  a:  =  2  ft. 


96  SCHOOL  ALGEBRA 

11.  x'-xix-hd)  =  x-{- 12.    13.  7(2  -  3x)  =  2(7"  -  Sx). 

12.  2a;  -  3(a:  -  3)  +  2  =  0.    i4.   3  -  2(3a:  +  2)  ==  7. 

15.  (x  -  8)  (a;  +  12)  -  (a:  +  1)  (x  -  6)  =  0. 

16.  5(a;  -  3)  -  7(6  -  a:)  +  3  =  24  -  3(8  -  x). 

17.  3(a;  -  1)  (a;  +  1)  =  a:(3a;  +  4). 

18.  4(a:  -  3)2  =  (2a:  +  1)1 

19.  8(a:  -  3)  -  (6  -  2a:)  =  2(a:  +  2)  -  5(5  -  a:). 

20.  5a:  -  (3a:  -  7)  -  {4  -  2a:  -  (6a:  -  3)}  =  10. 

21.  a:  +  2  -  [.T  -  8  -  2{8  -  3(5  -  x)  -  a-}]  =  0. 

22.  2a:(a:  -  5)  -  {x^  -\-  (3a:  -  2)  (1  -  a:)}  =  (2a:  -  4)^. 

23.  Sx''  +  13a:-2{a:2-3  [(a:-l)  (3  +  a:)-2(a:  +  2)2]}  =  3, 

24.  .25a:  -  2  =  .2a:  +  3.  26.  fa:  -f  6  =  ^a:  +  8. 

25.  1.6x  -  .7  =  1.5a:  -  .3.  27.   gx  -  |  =  |  -  fa:. 

28.  (.2a:  +  .2)  (.4a:  -  .3)  =  (.4a:  -  .4)  (.2a:  +  .3). 

29.  What  right  have  we  to  change  the  equation  3a:  =  15 —2a; 
xo  the  form  3x  -f  2a:  =  15? 

30.  If  2x  —  3  =  5,  what  right  have  we  to  transpose  the 
—  3,  and  to  write  the  equation  in  the  form  2a:  =  5  +  3? 

31.  What  is  the  advantage  in  being  able  to  add  the  same 
number  to  both  members  of  an  equation?  To  transpose  a 
term?  To  divide  both  members  of  an  equation  by  the  same 
number? 

32.  Determine  which  of  the  following  are  identities  and 
which  conditional  equations,  or  equations  proper: 

(1)  (x  +  3)  (a:  -  3)  =  a:2  -  9. 

(2)  (a:  +  3)  (x  -  3)  =  (a:  +  1)  (a:  -  2). 

(3)  (a:  +  2)  (a:  -  1)  =  a:2  +  a:  -  2. 

(4)  (a:  +  2)  (a:  -  1)  =  x\ 


EQUATIONS  97 

33.  Write  an  identity  and  an  equation  of  which  the  first 
members  are  the  same. 

34.  Prove  that  the  sum  of  any  three  consecutive  numbers 
equals  three  times  the  middle  one  of  the  numbers. 

SuG.    Let  the  three  numbers  be  indicated  by  n,  n  —  1,  and  n  —  2. 

35.  Find  a  similar  result  for  the  sum  of  five  consecutive 
numbers.    Of  seven  consecutive  numbers. 

36.  Prove  that  the  product  of  the  sum  and  difference  of 
any  two  numbers  is  equal  to  the  square  of  the  first,  minus 
the  square  of  the  second.    Illustrate  by  a  numerical  example. 

SuG.    Denote  the  two  numbers  by  a  and  b. 

37.  Prove  that  the  square  of  the  sum  of  any  two  numbers 
«^quals  the  square  of  the  first  number,  plus  twice  the  product 
of  the  two  numbers,  plus  the  square  of  the  second  number. 
Illustrate  by  a  numerical  example. 

38.  State  and  prove  a  similar  property  for  the  square  of 
the  difference   of  two  numbers. 

39.  Prove  that  if  the  sum  of  the  cubes  of  two  numbers  is 
divided  by  the  sum  of  the  numbers,  the  quotient  equals  the 
square  of  the  first  number,  minus  the  product  of  the  first  by 
the  second,  plus  the  square  of  the  second. 

40.  State  and  prove  a  similar  property  of  the  difference  of 
the  cubes  of  two.  numbers. 

Find  the  value  of  the  letter  in  each  of  the  following: 

41.  3a  -  2  =  7.  44.  24  =  12  -  3p. 

42.  5  -  26  =  1.  45.  3(2/  -  4)  =  5(2  -  y). 

43.  5(c  -  1)  =  12  -  c.         46.   r  -  3(r  -  1)  =  5. 

47.  In  ^  =  Iw,  if  A  =  42 J  and  I  =  8 J,  find  w, 

48.  If^  =  48.36  sq.  ft.  and  w  =  6.2  ft.,  find  I 


98  SCHOOL  ALGEBRA 

49.  Convert  each  of  the  two  preceding  examples  into  an 
example  concerning  areas. 

50.  If  V  =  Iwh,  and  V  =  504,  I  =  12,  and  ^  =  5,  find  w. 

51.  Convert  Ex.  50  into  an  example  concerning  volumes. 

52.  In  i  =  prt,  if  i  =  $27,  r  =  .05,  and  p  =  $240,  find  t. 

53.  Convert  Ex.  52  into  an  example  concerning  interest. 

54.  So  far  as  we  know,  who  first  used  an  equation  to  solve 
a  problem?  Give  this  first  problem  thus  solved,  and  tell  all 
you   know  about  the  document  in  which  it  was  found. 

(See  pp.  266  and  274.) 

55.  Form  an  equation  whose  root  is  2  and  which  contains 
four  terms. 

56.  Make  up  and  work  an  example  similar  to  Ex.  15.  To 
Ex.  31.    Ex.  46. 

57.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

72.  Solution  of  Problems.  In  solving  problems,  the  stu- 
dent will  find  it  necessary  to  study  each  problem  carefully 
by  itself,  as  no  rule  or  method  can  be  found  which  will  cover 
all  cases.  The  following  general  directions  will,  however,  be 
found  of  service: 

By  study  of  the  problem,  determine  what  are  the  unknown 
quantities  whose  values  are  to  be  obtained; 

Let  X  equal  one  of  these  unknown  quantities; 

State  in  terms  of  x  all  the  other  unknown  quantities  which  are 
either  to  be  determined  or  to  be  used  in  the  process  of  the  solution; 

Obtain  an  equation  by  the  use  of  a  principle  (such  as,  the 
whole  is  equal  to  the  sum  of  its  parts,  or  things  equal  to  the 
same  things  are  equal  to  each  other) ; 


EQUATIONS  99 

Solve  the  equation,  and  find  the  value  of  each  of  the  unknown 
quantities. 

In  solving  problems  it  is  especially  important  to  note  that 
we  let  X  =  a  definite  number ,  not  a  vague  quantity. 

Thus,  in  working  Ex.  1  of  Exercise  24 

we  do  not  let      x  =  A's  marbles, 
nor  X  =  what  A  has, 

but  let  X  =  number  of  marbles  A  has. 

73.  Checking  the  Solution  of  a  Written  Problem.  The 
best  way  of  checking  the  result  obtained  by  solving  a  prob- 
lem is  to  observe  whether  the  result  obtained  satisfies  the 
conditions  as  originally  stated  in  the  language  of  the  problem. 
(This  method  is  better  than  that  used  in  checking  the  example 
in  Art.  46,  p.  53.) 

Thus,  to  check  Ex.  18,  p.  54 :  after  the  answers  9  and  4  have  been 
obtained,  we  note  that  the  difference  of  9  and  4  is  5,  and  that  the 
sum  of  9  and  4  is  13.  9  and  4  thus  satisfy  the  original  conditions  ot 
the  problem. 

What  is  the  advantage  in  this  method  of  checking  the  solu- 
tion of  a  problem? 

EXERCISE  24. 

#  i^       Oral 

1.  A  has  X  marbles,  and  B  has  twice  as  many.  How  many  has  B? 
How  many  have  both? 

2.  There  are  100  pupils  in  a  school,  of  which  x  are  boys.  How 
many  are  girls? 

3.  If  I  have  x  dollars,  and  you  have  three  dollars  more  than 
twice  as  many,  how  many  have  you?    How  many  have  we  together? 

4.  Two  boys  together  solved  a  examples.  One  did  x  examples. 
How  many  did  the  other  solve? 

5.  The  difference  between  two  numbers  is  15,  and  the  less  is  x. 
What  is  the  greater?    What  is  their  sum? 


100  SCHOOL  ALGEBRA 

6.  If  n  is  a  whole  number,  what  is  the  next  larger  number?    The 

next  less? 

7.  Write  three  consecutive  numbers,  the  least  being  x.      Write 
them  if  the  greatest  is  y. 

8.  John  has  x  dollars,  and  James  has  seven  dollars  less  than 
three  times  as  many.    How  many  has  James? 

9.  If  I  am  a:  years  old  now,  how  old  was  I  ten  years  ago?    a  years 
ago?    How  old  will  I  be  in  c  years? 

10.  A  man  bought  a  horse  for  x  dollars,  and  sold  it  so  as  to  gain 
a  dollars.    What  did  he  receive  for  it? 

11.  A  man  sold  a  horse  for  $200,  and  lost  x  doUars.    What  did 
the  horse  cost? 

12.  If  a  yard  of  cloth  cost  m  dollars,  what  will  x  yards  cost? 

13.  A  boy  rides  a  miles  an  hour;  how  far  will  he  ride  in  c  hours? 

14.  A  bicycUst  rides  x  yards  in  y  seconds.    How  far  will  he  ride 
in  one  second?    In  n  seconds? 

15.  In  how  many  hours  can  a  boy  walk  x  miles  at  a  miles  an  hour? 

16.  A  man  has  a  dollars  and  h  quarters.    How  many  cents  has  he? 

17.  How  many  dimes  in  x  dollars  and  y  half-dollars? 

18.  I  have  x  dollars  in  my  purse  and  y  dimes  in  my  pocket.    If 
I  give  away  fifty  cents,  how  much  have  I  remaining? 

19.  By  how  much  does  30  exceed  xt 

20.  What  number  is  40  less  than  a;?    What  number  is  x  less 
than  40? 

21.  What  number  exceeds  x  by  a?   What  number  exceeds  a  by  x? 

22.  By  how  Imuch  does  a  -\-h  exceed  x'i 

23.  How  much  did  a  girl  have  left  if  she  had  $5  and  spent  15ff? 
If  she  had  a  dollars  and  spent  h  cents? 

24.  A  boy  had  a  dollars,  received  h  cents,  and  then  spent  c  cents. 
How  many  cents  did  he  have  left? 

25.  What  is  the  interest  on  a  dollars  at  h  per  cent  for  c  years? 

26.  Express  algebraically  the  following  statement:  a  divided  by 
6  gives  c  as  a  quotient  and  rf  as  a  remainder. 


I 


EQUATIONS  101 


27.  A  man  having  x  hours  at  his  disposal,  rode  a  hours  at  the 
rate  of  8  miles  an  hour,  and  walked  the  rest  of  his  time  at  the  rate 
of  3  miles  an  hour.    How  far  did  he  ride?    How  far  did  he  walk? 

EXERCISE  25 

l^ft  1.  Separate  $84  into  two  parts  such  that  one  part  is  three 
^mes  as  large  as  the  other. 

2.  Separate  $84  into  two  parts  such  that  one  part  exceeds 
the  other  by  $12. 

3.  Separate  $84  into  three  parts  such  that  the  first  part 
is  twice  as  large  as  the  second,  and  the  second  part  is  twice  as 
large  as  the  third. 

4.  A  boy  has  three  times  as  many  marbles  as  his  brother, 
and  together  they  have  48;  how  many  has  each? 

5.  A  and  B  pay  together  $100  in  taxes;  if  A  pays  $22  more 
than  B,  what  does  each  pay? 

6.  Two  boys  made  $67.50  one  summer  by  taking  passengers 
on  a  launch.  The  boy  who  owned  the  launch  received  twice 
as  large  a  share  of  the  profits  as  the  other  boy.  How  much 
did  each  receive? 

7.  How  many  grains  of  gold  are  there  in  a  gold  dollar,  if 
the  gold  dollar  weighs  25.8  grains  and  9  parts  of  the  dollar 
are  gold  and  1  part  copper? 

8.  A  ball  nine  has  played  64  games  and  won  12  more  than 
it  has  lost.    How  many  games  has  it  won? 

9.  A  man  left  $21,000  to  his  wife  and  four  daughters.  If 
the  wife  received  three  times  as  much  as  each  daughter,  how 
much  did  each  receive? 

10.  If  he  had  left  $21,000  so  that  the  wife  received  $10,000 
more  than  each  daughter,  how  much  would  each  have 
received? 


102  SCHOOL  ALGEBRA 

11.  A  cubic  foot  of  water  and  a  cubic  foot  of  alcohol  to- 
gether weigh  112.5  lb.  The  alcohol  weighs  5  as  much  as  the 
water.    What  is  the  weight  of  a  cubic  foot  of  each? 

12.  Find  three  consecutive  numbers  whose  sum  is  63. 

13.  In  a  certain  grade  of  milk  the  other  solids  equal  three 
times  the  weight  of  the  butter  fat,  and  the  liquid  part  of  the 
milk  weighs  7  times  as  much  as  the  solids.  How  many  pounds 
of  butter  fat  in  4800  lb.  of  milk? 

14.  The  difference  of  the  squares  of  two  consecutive  num- 
bers is  43.    Find  the  numbers. 

15.  At  a  certain  date  the  record  time  for  the  quarter-mile 
run  was  47  seconds,  and  5  times  the  record  time  for  the  100- 
yard  dash  exceeded  the  record  time  for  the  quarter-mile  by 
1  second.  Find  the  record  time  for  the  100-yard  dash  at  this 
date. 

16.  The  difference  of  two  numbers  is  13  and  their  sum  is 
35.    Find  the  numbers. 

17.  John  solved  a  certain  number  of  examples,  and  William 
did  12  less  than  twice  as  many.  Together  they  solved  96» 
How  many  did  each  solve? 

18.  Three  boys  earned  together  $98.  If  the  second  earned 
$11  more  than  the  first,  and  the  third  $28  less  than  the  other 
two  together,  how  many  dollars  did  each  earn? 

19.  The  sum  of  two  numbers  is  92,  and  the  larger  is  3  less 
than  four  times  the  less.    Find  the  numbers. 

20.  The  sum  of  three  numbers  is  50.  The  first  is  twice 
the  second,  and  the  third  is  16  less  than  three  times  the  second. 
Find  the  r umbers. 


I 


EQUATIONS  103 


21.  A  farmer  paid  $94  for  a  horse  and  cow.  What  did 
each  cost,  if  the  horse  cost  $13  more  than  twice  as  much  as  the 
cow?    X  '\    '\  "^  -^  ■■■'^  -    ?^/ 

22.  Ex.  1  (p.  95)  might  be  stated  as  a  problem  concerning 
an  unknown  number,  thus:  Twice  a  certain  number  equals  15 
less  three  times  the  number.    Find  the  number. 

In  like  manner,  convert  Ex.  2  (p.  95)  into  a  problem  con- 
cerning an  unknown  number.    Also  Ex.  3.    Ex.  8. 

23.  In  reducing  iron  ore  in  a  furnace,  7  times  as  many  car- 
loads of,  coke  as  of  limestone  are  used,  and  8  times  as  many 
carloads  of  iron  ore  as  of  limestone.  If  800  carloads  in  all  are 
used  on  a  certain  day,  how  many  carloads  of  each  is  this? 

24.  One  side  of  a  triangle  is  twice  as  long  as  the  shortest 
side.  The  third  side  exceeds  the  length  of  the  shortest  side 
by  12  yards.  If  the  perimeter  of  the  triangle  is  360  yards,  find 
each  side. 

25.  A  man  spent  $3.24  for  coffee  and  sugar,  buying  the 
same  number  of  pounds  of  each.  If  the  sugar  cost  5  cents  a 
pound  and  the  coffee  22  cents,  how  many  pounds  of  each  did 
he  buy? 

26.  The  distance  from  New  York  to  Chicago  is  912  miles. 
If  this  is  24  miles  less  than  four  times  the  distance  from 
New  York  to  Boston,  find  the  latter  distance. 

27.  On  a  certain  railroad  in  a  given  year  the  receipts  per 
mile  were  $3085.  If  the  receipts  per  mile  for  freight  exceeded 
those  for  passengers  by  $265,  find  the  receipts  per  mile  from 
each  of  these  sources. 

28.  A  man  left  $64,000  to  his  wife,  daughter,  and  niece. 
To  his  daughter  he  left  $4000  more  than  to  his  niece,  and  to 
his  wife  $8000  more  than  to  his  daughter  and  niece  together. 
How  much  did  he  leave  to  each? 


104 


SCHOOL  ALGEBRA 


a;  +  5 


29.  Find  the  number  whose  double  exceeds  24  by  6. 

30.  The  perimeter  of  a  given  rectangle  is  26  feet,  and  the 
length  of  the  rectangle  exceeds  the 
width  by  5  feet.  Find  the  dimen- 
sions of  the  rectangle. 

31.  The  perimeter  of  a  given 
rectangle  is  18  yards,  and  the 
length  exceeds  the  width  by  3  ft.  Find  the  dimensions. 
Make  up  and  work  a  similar  example  for  yourself. 

32".  The  length  of  a  rectangle  exceeds  a  side  of  a  given 
square  by  3  inches  and  the  width  of  the  rectangle  is  2  inches 
less  than  a  side  of  the 
square.  If  the  area  of 
the  rectangle  equals  the  ^ 
area  of  the  square,  find 
a  side  of  the  square. 

SuG.  Denote  the  sides  of  the  square  and  rectangle  as  in  the 
diagram. 

Since  the  areas  of  the  two  figures  are  equal, 

x^  =  (x  +  3)  (x  -  2),  etc. 
In  working  Exs.  33-38,  draw  a  diagram  for  each  example. 

33.  The  length  of  a  rectangle  exceeds  a  side  of  a  given 
square  by  8  ft.  and  the  width  of  the  rectangle  is  4  ft.  less  than 
a  side  of  the  square.  If  the  area  of  the  rectangle  equals  the 
area  of  the  square,  find  a  side  of  the  square. 

34.  If  one  side  of  a  square  is  increased  by  4  yd.,  and  an 
adjacent  side  by  3  yd.,  a  rectangle  is  formed  whose  area  ex- 
ceeds that  of  the  square  by  47  sq.  yd.  Find  a  side  of  the 
square. 

35.  The  perimeter  of  a  rectangle  is  120  ft.,  and  the  rec- 
tangle is  twice  as  long  as  it  is  wide.    Find  its  dimensions. 


ic  +  S 


EQUATIONS  105 

36.  A  certain  rectangle  is  three  times  as  long  as  it  is  wide. 
If  20  ft.  is  added  to  its  length  and  10  ft.  is  deducted  from 
its  width,  the  area  is  diminished  by  400  sq.  ft.     Find  the 

i    dimensions  of  the  rectangle. 

IV  37.  A  rectangle  is  5  ft.  longer  than  it  is  wide.  If  its  length 
is  increased  by  4  ft.,  and  its  width  by  3  ft.,  its  area  is  in- 
creased by  76  sq.  ft.  Find  the  dimensions  of  the  rectangle. 
38.  A  rectangle  is  4  in.  longer  than  it  is  wide.  If  its  length 
is  increased  by  4  in.,  and  its  width  diminished  by  2  in.,  its  area 
remains  unchanged.    Find  the  dimensions  of  the  rectangle. 

139.  Make  up  and  work  an  example  similar  to  Ex.  38. 
40.  A  tennis  court  is  42  ft.  longer  than  it  is  wide.    If  a 
argin  of  15  ft.  on  each  end  and  of  10  ft.  on  each  side  is 
added,  the  area  of  the  court  is  increased  by  3240  sq.  ft. 
Find  the  dimensions  of  the  court. 

41.  The  length  of  a  football  field  exceeds  its  width  by  140 
ft.  If  a  margin  of  20  ft.  is  added  on  each  side  and  end  of  the 
field,  the  area  is  increased  by  20,000  sq.  ft.  Find  the  dimen- 
sions of  the  field. 

42.  A  boy  is  three  times  as  old  as  his  brother.  Five  years 
hence  he  will  be  only  twice  as  old.  Find  the  present  age  of 
each. 

43.  A  man  is  twice  as  old  as  his  brother.  Five  years  ago 
he  was  three  times  as  old.  Find  the  age  of  each  at  the  present 
time. 

44.  How  many  pounds  of  coffee  at  30^  a  pound  must  be 
mixed  with  12  pounds  of  coffee  at  20f^  a  pound  to  make  a 
mixture  worth  24^  a  pound? 

45.  How  many  pounds  of  tea  at  60^  a  pound  must  be 
mixed  with  25  lb.  of  tea  at  40^  a  pound,  to  make  a  mixture 
worth  4i5^  a  pound? 


106  SCHOOL  ALGEBRA 

46.  Make  up  and  work  an  example  similar  to  Ex.  45. 

47.  Find  five  consecutive  numbers  whose  sum  shall  be  3 
less  than  six  times  the  least. 

48.  Find  three  consecutive  odd  numbers  whose  sum  is  63. 

49.  A  telegram  at  a  25-2  rate  cost  47  cents.  How  many 
words  were  in  the  telegram? 

SuG.  A  25-2  rate  means  a  cost  of  25  cents  for  the  first  10  words 
and  2  cents  for  each  additional  word. 

50.  Make  up  and  work  an  example,  similar  to  Ex.  49,  con- 
cerning a  telegram  sent  at  a  40-3  rate. 

51.  A  talk  over  a  long  distance  telephone  at  a  50-7  rate 
cost  85f^.    How  many  minutes  did  the  talk  last? 

SuG.  A  50-7  rate  over  a  long  distance  telephone  means  a  cost 
of  50  cents  for  the  first  3  minutes  and  7  cents  for  each  additional 
minute. 

52.  A  rectangle  is  8  ft.  longer  than  it  is  wide  and  the  pe- 
rimeter is  120  ft.    Find  the  dimensions  of  the  rectangle. 

53.  If  5  is  subtracted  from  a  certain  number  and  the  difTer- 
ence  is  subtracted  from  115,  the  result  equals  three  times  the 
given  number.    Find  the  number. 

54.  If  J  is  added  to  double  a  certain  fraction,  the  result  ii^ 
the  same  as  if  |  had  been  subtracted  from  three  times  the 
fraction.    Find  the  fraction. 

55.  What  number  subtracted  from  100  gives  a  result  equal 
to  the  sum  of  14  and  the  number? 

56.  Find  the  number  which  exceeds  12  by  as  much  as 
three  times  the  number  exceeds  24. 

57.  Find  five  consecutive  numbers  such  that  the  last  is 
twice  the  first. 


EQUATIONS  107 

58.  Find  two  consecutive  integers  such  that  the  first  plus 
5  times  the  second  equals  53. 

59.  A  man  is  48  years  old  and  his  son  is  18.  How  many 
years  ago  was  the  father  four  times  as  old  as  the  son?  Also 
how  many  years  hence  will  the  father  be  twice  as  old  as  the 
son? 

60.  Find  two  numbers  such  that  their  difference  is  20,  and 
one  is  four  times  as  large  as  the  other. 

61.  The  length  of  a  single  tennis  court  exceeds  the  width 
by  51  ft.  If  the  width  is  increased  by  9  ft.,  we  have  a  double 
court,  the  area  of  which  exceeds  that  of  the  single  court  by 
702  sq.  ft.    Find  the  dimensions  of  each  court. 

62.  A  boy  sold  a  certain  number  of  newspapers  on  Monday, 
twice  as  many  on  Tuesday,  on  Wednesday  5  more  than  on 
]\Ionday,  and  on  Thursday  7  less  than  on  Tuesday.  If  he 
sold  310  newspapers  on  the  four  days,  how  many  did  he  sell 
on  each  of  the  days? 

63.  Twenty-five  men  agreed  to  pay  equal  amounts  in 
raising  a  certain  sum  of  money.  Five  of  them  failed  to  pay 
their  subscriptions,  and  as  a  result  each  of  the  other  twenty 
had  to  pay  one  dollar  more.  How  much  did  each  man  sub- 
scribe originally? 

64.  A  boy  starts  from  a  certain  place  and  walks  at  the 
rate  of  3  miles  an  hour.  Three  hours  later  another  boy  starts 
after  the  first  boy  and  travels  on  a  bicycle  at  the  rate  of  6 
miles  an  hour.  How  many  hours  will  it  be  before  the  second 
boy  overtakes  the  first?    (Draw  a  diagram.) 

65.  If  the  boys  had  traveled  in  opposite  directions,  how 
many  hours  after  the  second  boy  started  would  it  have  been 
before  they  were  81  miles  apart? 


108  SCHOOL  ALGEBRA 

66.  A  boy  was  engaged  to  work  50  days  at  75^  per  day  for 
the  days  he  worked,  and  to  forfeit  25^!^  every  day  he  was  idle. 
On  settlement  he  received  $25.50;  how  many  days  did  he 
work? 

67.  Which  of  the  above  problems  belong  to,  or  are  varia- 
tions of.  Type  I?    Of  Type  II?    III? 

68.  How  many  examples  in  Exercise  15  (p.  60)  can  you 
now  work  at  sight? 


CHAPTER  VII 
ABBREVIATED   MULTIPLICATION  AND   DIVISION 

p  Abbreviated  Multiplication 

74.  Utility  of  Abbreviated  Multiplication.  In  certain 
cases  of  multiplication,  by  observing  the  character  of  the 
expressions  to  be  multiplied,  it  is  possible  to  write  out  the 
product  at  once,  without  the  labor  of  the  actual  multiplica- 
tion. This  is  true  of  almost  all  the  multiplication  of  binomials, 
and  that  of  many  trinomials,  and  by  the  use  of  the  abbre- 
viated methods  at  least  three  fourths  of  the  labor  of  multi- 
plication in  such  cases  may  be  saved.  The  student  should 
therefore  master  these  short  methods  as  thoroughly  as  the 
multiplication  table  in  arithmetic. 

75.  I.   Square  of  the  Sum  of  Two  Quantities. 

Let  a  +  6  be  the  sum  of  any  two  algebraic  quantities. 
By  actual  multiplication,  a  -{-  b 

a  +  h 

a^  4-  ab 

a^  +  2ab  +  b^  Product 
Or,  in  brief,  (a  +  bf  =  a^  +  2ab  +  ¥, 

which,  stated  in  general  language,  is  the  rule: 

The  square  of  the  sum  of  two  quantities  equals  the  square  of 
the  first,  plu^  twice  the  product  of  the  first  by  the  second,  plus  the 
square  of  the  second. 

109 


110       '  SCHOOL  ALGEBRA 

Ex.  1.    (2x  +  3yy  =  4a:2  +  12xy  +  V  Prodmjt 
Ex.  2.    1042  =  (100  +  4)2  =  1002  H-  8  X  100  +  42 
=  10,000  +  800  +  16  =  10816  Ans. 

76.  n.   Square  of  the  DiiFerenee  of  Two  Quantities. 
By  actual  multiplication,  a  —  b 

a  —  b 

€?  —  ab 

-ab    -\-b' 

G?  -  2ab  -f-  62  Product 

Or,  in  brief,  (a  -  bf  =  a"  -  2ab  +  b\ 

which,  stated  in  general  language,  is  the  rule: 

The  square  of  the  difference  of  two  quantities  equals  the  square 

of  the  first,  minu^  twice  the  product  of  the  first  by  the  second, 

plus  the  square  of  the  second. 

Ex.  1.    (2x-  3yf  =  4a:2  -  12xy  +  9y^  Product 

Ex.  2.    [{x  +  2y)-3f  =  (x  +  2yy-  10(a:  +  2y)  +  25 

=  a:2  +  4.T2/  +  4?/2  -  10.r  -  2O2/  +  25 

Product 

To  check  the  work  of  Ex.  2,  let  a;  =  2,  y  =  1. 
Then      [{x^  +  2y)  -  5]2  =  (4  +  2  -  5)^  =  (1)2  =  1 
Also 
x^  +4xy  +^y^  -  lOx  -20?/ +25  =4+8+4  -20-20  +25  =  1. 

EXERCISE  26 

Write  by  inspection  the  value  of  each  of  the  following  and 
check  each  result: 

1.  (n  +  yy  5.    (5a: +1)2 

2.  {c-xY  6.    (0:2+1)2 

3.  (2x  -  yf  7.    {x  —  2/2)2 

4.  (3a:  -  2yf  8.  (1  -  Iff 


I 


ABBREVIATED   MULTIPLICATION  111 

9.  (3x4  ^  5^)2  13  (1,5^  _  .02)2 

10.  (GarV  -  lli/V)2  19.  [(a  +  6)  +  4]^ 

11.  (5a:»  -  3i/"z-)2  20.  [(a  +  6)  -  3]2 

12.  (4a:32^s2"  +  V")2  21.  [(a  +  h)  +  c]^ 

13.  (|a:^  +  f2/)'  22.  [(2a  -  .t)  +  37/]2 

14.  (fa6  -  |x2)2  23.  [3  +  (a  +  &)]' 
-^05.  (.2x  +  2yf  24.  [5a  -  (a:  +  2/)]' 

16.  (.3a  +  .0462)2  25.  [2a2  -  (6  -  2c) ]2 

17.  (.02a:  -  myf  26.  [{x  +  2/)  -  («  +  &)P 

27.  Find  the  value  of  998^  by  multiplying  998  by  itself. 

This  product  might  also  have  been  obtained  in  the  following 

U^ay: 

9982  =  (1000  -2)2  =  [10002  -  2  x  2  X  1000  +  22] 
=  1,000,000  -  4000  +  4  . 
=  996,004 

After  practice  the  part  of  the  work  in  the  brackets  may  be  omitted. 
Compare  the  amount  of  work  in  the  two  processes  of  finding  th? 
value  of  9982. 

By  the  short  method  obtain  the  value  of: 

28.  9992  31.  512  34    9962 

29.  9972  32.    10032  35.    99972 

30.  99982  33.  972  36.   (99.2)2 

37.  Make  up  and  work  an  example  similar  to  Ex.  19.    To 
Ex.29.    Ex.36. 

38.  How  many  of  the  examples  in  this  Exercise  can  you 
answer  orally? 


112  SCHOOL  ALGEBRA 

77.  III.  Product  of  the  Sum  and  Difference  of  Two 
Quantities. 

By  actual  multiplication,  a  -\-  b 

a  —  b 
a^  +  ab 

-ab-W 
q2  _  52  Product 

Or,  in  brief,  (a  +  b)  {a  -  b)  =  a^  -  b\ 

which,  stated  in  general  language,  is  the  rule: 

The  product  of  the  sum  and  difference  of  two  quantities  equals 
the  square  of  the  first  minu^s  the  square  of  the  second. 

Ex.  1.   (2x  H-  32/)  {2x  -  3y)  =  4a^  -  9y^  Product 

Ex.  2.  Multiply  x  -^  {a  -^  b)  hy  x  -  {a  -{-  b). 

We  have 

lx  +  {a+  b)]  [x  -(a+  b)]  =x^  -{a+  by,  by  IIL 

=  a;2  _  (a2  +  2ab  +  ¥),  by  I. 

=  a;2  -  a2  -  2ab  -  b^  Product 
Let  the  pupil  check  the  work. 

It  is  frequently  necessary  to  re-group  the  terms  of  trino- 
mials in  order  that  the  multiplication  may  be  performed  by 
the  above  method. 

Ex.  3.  Multiply  x  +  y  —  zhyx  —  y  +  z. 

{x  +y  -  z)  (x  -y  +z)  =  [x  -{■  (y  -  z)][x  -  (y  -  z)] 
=  x^  -{y  -  z)\  by  III. 
=  a^2  _  (^2  _  2yz  +  z^),  by  II. 
=  a:2  -  2/2  +  2yz  -  z^    Product 
Let  the  pupil  check  the  work. 

EXERCISE  27 

Write  by  inspection  the  value  of  each  of  the  following 
products,  and  check  the  work  for  each  result: 

1.  (x  +  z)  (x  -  z)  3.    (3a;  -  y)  (Sx  +  y) 

2.  (2/-3)(y  +  3)  4.   (7a:  +  41/)  (7a:  -  42/) 


I 


ABBREVIATED  MULTIPLICATION  113 


5.  (x^  -2)(x'-h  2)  9.   (ia  +  ib)  iia  -  J6) 

6.  (ax'  -  b'y)  (ax'  +  b'y)       lo.   (2iar  -  iy)  (2ix  +  iy) 

7.  (1  -  lla:^)  (1  _^  iia:^)         ^^^    (.2^  ^  3^)  ^2x  -  .S^/) 

8.  (2a;»  +  5!/-)  (2a:"  -  52/-)     12.    (.05a2-.363)(.05a2+.3fe3) 

13.  (fa:  +  .7y)  (ix  -  ,7y) 

14.  (a»+^  +  ^6-1)  (a"+i  -  i6"-i) 

15.  [(a  +  6)  +  3]  [(a  +  6)  -  3] 

16.  [(a:  +  2/)  +  a]  [(x  +  2/)  -  a] 

17.  [(2a:  -  1)  +  2/]  [(2a;  -  1)  -  2/] 

18.  [4  +  (a:  +  1)]  [4  -  (a:  +  1)] 

19.  [2a:  +  (32/  -  5)]  [2a:  -  (32/  -  5)] 

20.  (a  +  6  +  3)  (a  +  6  -  3) 

21.  (a:  +  2/  +  a)  (a:  +  2/  -  a) 

22.  (4  +  a:  +  1)  (4  -  a:  -  1) 

23.  (2a:  +  32/  -  5)  (2a:  -  32/  +  5) 

24.  (4  +  a:  +  2/)  (4  -  a:  -  2/) 

25.  (a:2  +  3a:  +  2)  (a:^  +  3a:  -  2) 

26.  (a  4-  6  +  3a:)  (a  +  6  -  3a:) 

27.  (a  +  6  -  3a:)  (a  -  6  +  3a:) 

28.  (a:2  -  a:2/  +  y^)  (x^  +  xy  -{-  y^) 

29.  (a''  +  a-{-  1)  (a2  -  a  +  1) 

30.  (2a:2  -  3a:  -  5)  (2a:2  +  3a:  -  5) 

31.  (2a:2  4-  5xy  -  if)  (2^?  -  bxy  -  y^) 

32.  (x'  +  xy  -  2/2)  (x'  -  xy  -  y^) 

33.  l(a  +  b)-(c-  1)]  [(a  +  6)  +  (c  -  1)] 

34.  [(a:2  +  f)  +  (:r22/2  +  1)]  [(x'  +  2/^)  -  (0:2^2  _^  i)j 

35.  (a;  +  1/  +  z  +  1)  (a:  +  2/  -  z  -  1) 


U4  SCHOOL  ALGEBRA 

36.  Work  Ex.  16  in  full  (see  Art.  54,  p.  65).    How  much 
of  this  labor  is  saved  by  the  short  method  of  multiplication? 

37.  Make  up  and  work  an  example  similar  to  Ex.  36. 

38.  Multiply  93  by  87.    This  product  may  also  be  obtained 
thus: 

93  X  87  =  (90  +  3)  (90  -  3) 
=  8100  -  9  =  8091 
Compare  the  amount  of  work  in  the  two  processes. 

39.  Make  up  and  work  an  example  similar  to  Ex.  38. 
Find  the  value  of  each  of  the  following  in  the  short  way: 

40.  92  X  88  43.   1005  X  995 

41.  103  X  97  44.   1032  -  972 

42.  105  X  95  45.   (17.31)2  -  (2.69)2 
Find  in  the  shortest  way: 

46.  The  area  of  a  rectangle  102  ft.  long  and  98  ft.  wide. 

47.  The  cost  of  32  doz.  eggs  at  28f^  per  dozen. 

48.  The  cost  of  67  yd.  of  cloth  at  73^  a  yard. 

49.  Make  up  and  work  two  examples  similar  to  Exs.  47^8. 

50.  Work  Ex.  40  in  full.    How  much  of  this  labor  is  saved 
by  using  the  short  method  of  multiplication? 

Write  at  sight  the  product  for  each  of  the  following  miscel- 
laneous examples: 


51. 

(x  +  2a)2 

58. 

l(x  +  2y)  +  5]2 

52. 

(x  +  2a)  {x  -  2a) 

59. 

(x^2y-\-5){x-h2y-5) 

53. 

(x  -  2a)2 

60. 

(.3a:  +  .5y)  (.3a:  -  .5y) 

54. 

(3a:  -  1)  (3a:  +  1) 

61. 

9982 

55. 

(3a:  -  1)2 

62. 

998  X  1002 

56. 

(3a2  -  26^)2 

63. 

972 

57. 

(3a2  -  2¥)  (3a2  +  2¥ 

')     64. 

97  X  103 

I 

■  ABBREVIATED  MULTIPLICATION  115 

^K   65.  Make  up  and  work  an  example  in  each  principal  form 
^Bf  abbreviated  multiplication  studied  thus  far. 

66.  How  many  of  the  examples  in  this  Exercise  can  you 
answer  orally? 

78.  IV.   Square  of  any  Polynomial. 
By  actual  multiplication, 
a  -\-b      +    c 
g  4-  6      +    c 
a^  -^  ab    -}-    ac 

+  ab  +62+    6c 

-^    ac  +    6c  +  c^ 


a^  +  2ab  +  2ac  +  6^  +  26c  +  c^  Product 
Or,  in  brief,  (a  +  6  +  c)^  =  a^  +  6^  +  c^  +  2ab  +  2ac  +  26c. 
In  like  manner  we  obtain 

(a  +  6  +  c  +  6^)2  =  a2  +  62  +  c2  +  (Z2  4-  2a6  +  2ac  +  2ad 

+  26c  +  2bd  +  2cd 
Or,  in  general. 

The  square  of  any  polynomial  equals  the  sum  of  the  squares 
of  the  terms  plu^  twice  the  product  of  each  term  by  each  term 
which  follows  it. 

It  is  often  useful  to  indicate  the  order  in  which  the  products  of 
the  terms  are  taken  as  shown  in  the  following  diagram.  (If  the 
curved  Unes  joining  the  terms  are  drawn  as  each  product  is  taken, 
the  numbers  on  these  lines  may  be  omitted.) 


Ex.   (a  -  26  +  c  -~3x)2  =  a^  +  W  +  c^  +  9x2  -  4a6  +  2ac  -  6aa; 


-46c+  126a;  -  Qcx. 
Let  the  pupil  check  the  work. 


116  SCHOOL  ALGEBRA 

EXERCISE  28 

Find  in  the  shortest  way  the  value  of  the  following  and 
cheek  each  result: 

1.  (2a;  +  2/  +  1)'  8.  {2c?  +  5a  -  3)^ 

2.  (a:  -  2?/  +  2zf  9.  {x  -  y  ■\- z  -  Xf 

3.  (3a;  -2y-  5)^  lo.  {2x-\-Zy-Az-  hf 

4.  (2a  -  6  +  3c)2  11.  (3a;3  _  4^  _^  ^  _  2)2 

5.  (a; -21/ -32)2  ^3.  (|a;2  -  f  a:  +  5)^ 

6.  (4a;  +  3^  -  l)^  13.  {\x^  -  \x^  +  f  a?  +  6)^ 

7.  (a;2  -  a;  +  1)^  14.  (.2a  +  .3&  -  .hcf 

15.  Expand  (2a  —  36  +  c  —  4(Z)2  by  multiplying  in  full. 
Now  obtain  the  same  result  by  the  method  of  Art.  78,  p.  115. 
About  how  much  of  the  work  of  multiplication  is  saved  by 
the  latter  method? 

16.  Make  up  and  work  an  example  similar  to  Ex.  15. 
Write  at  sight  the  product  for  each  of  the  following  mis- 
cellaneous examples: 

17.  (2a -36)2  21.   (3a;2  + 2/3)2 

18.  (2a  +  36)  (2a  -  36)       22.   (3a;2  -  ^ff 

19.  (2  +  36)2  23.   (a; +  22/- 3)  (a: +  22/ +  3) 

20.  (3a;2  -  2/3)  (3^  ^  2/3)       24.   (a;  +  22/ -  3)2 

25.  (4a;  +  Ja-f)2 

26.  (4a;  +  ia-f)(4a;  +  ia  +  f) 

27.  (a;"+^  -  3x"-V)2 

28.  (a;"+i  -  3a;"-V)  (a;"+^  +  3a;"-V) 

29.  (.02x2  -  .3a;  +  .5)2 


I 


ABBREVIATED  MULTIPLICATION  117 


30.  Make  up  and  work  an  example  in  each  principal  form 
of  abbreviated  multiplication  studied  thus  far. 

79.   V.  Product  of  Two  Binomials  of  the  Form  a?  -f  a, 
a?  +  6. 

By  actual  multiplication, 

X  -h  5  X  —  by  x-\-  a 

g;  +3  a;  +  3i/  x  +  h 

x^  -\-  bx  oP-  —  5xy  x^  +  ax 
+  3a;  +  15            +  ^xy  -  15y^  -\- hx  +  ab 


r^  +  8a;  +  15        x""  -  2xy  -  152/^        x^  +  (a  +  h)x  +  ah 

By  comparing  each  pair  of  binomials  with  their  product, 
we  observe  the  following  relation: 

The  product  of  two  binomiab  of  the  form  x  -{-  a  and  x  -{-  b 
consists  of  three  terms: 

The  first  term  is  the  square  of  the  first  term  of  the  binomials; 

The  last  term  is  the  product  of  the  second  terms  of  the 
binomials; 

The  middle  term  consists  of  the  first  term  of  the  binomials 
with  a  coefficient  equal  to  the  algebraic  sum  of  the  second  terms 
of  the  binomials. 

Ex.  1.    Multiply  a;  -  8  by  a;  +  7. 

-8+7  =  - 1.        -8x7  =  - 56. 
.*.  (x  -S)  ix  +  7)  =x^  -X  -66  Product 

Ex.  2.    Multiply  (a;  -  6a)  (x  -  5a). 

(-  6a)  +  (-  5a)  =  -  11a.         (-  6a)  X  (-  5a)  =  +  SOa\ 
,'.{x  -6a)  {x  -  5a)  =  x^  -  Uax  -{-  SOa^  Product 

Ex.  3.    Multiply  x  +  y  +  Qhyx-^y-2. 

(x+y  +6){x+y  -2)  =l(x-{-y)  +6]  [(x  +  y)  -  2] 

^  {X  +  yY  +  4(rc  -^  y)  -  12  Product 


118  SCHOOL  ALGEBRA 

EXERCISE  29 

Write  the  product  for  each  of  the  following  and  check  each 
result: 

1.  {x  +  2){x  +  5)  10.  {x  +  .2)  {x  +  .5) 

2.  {x  "  5)  (a:  -  3)  li.  {x  +  |)  {x  +  \) 

3.  {x  -  7)  (a:  +  4)  12.  {x  +  .02)  {x  +  5) 

4.  (x  -  4)  (a:  +  8)  13.  (a  +  .02)  (a  +  .5) 

5.  (a:  +  l)(a:-7)  14.  {x  -  h)  {x  +  \) 

6.  {x^-2){x'-Z)  15.  (a  +  f)(a-i) 

7.  (a:2  +  3)  (a:2  +  1)  le.  {ah  +  x)  {ah  +  3a;) 

8.  {a  +  3a:)  (a  -  lOx)  17.  (a6  +  a:)  {ah  -  3x) 

9.  (a:  -  7y)  {x  +  2/)  18.  {xy  -  7z^)  {xy  +  32^) 

19.  (a;"  +  5)  (a:"  -  5) 

20.  [{x  +  2/)  +  3]  [(a:  +  ?/)  +  5] 

21.  [(a:  +  2/)  -  3]  [(a:  +  2/)  +  5] 

22.  (a;  +  2/  -  3)  (a:  +  2/  +  5) 

23.  (a  +  26  +  5)  (a  +  26  +  3) 

24.  {2x  +  3?/  +  3a)  (2a:  +  3?/  -  5a) 

25.  (a:  +  a  +  6)  (a:  —  a  —  6) 

26.  (2a:  +  a  +  36)  (2a:  -  a  -  36) 

27.  (2a:  +  a  -  36)  (2a:  -  a  +  36) 

28.  Find  the  product  of  a:  +  2/  +  ^  and  x  -{■  y  —  ?>  by 
multiplying  in  full.  Then  find  the  same  product  by  the 
method  of  Art.  79.  About  how  much  of  the  work  of  multi- 
plication is  saved  by  use  of  the  latter  method? 

2,9.  Make  up  and  work  an  example  similar  to  Ex.  27. 


I 


ABBREVIATED   MULTIPLICATION  119 


30.  A  building  lot  is  167  ft.  wide  and  213  ft.  deep.  If  the 
width  and  depth  of  the  lot  are  each  increased  by  1  foot,  find 
the  increase  in  area  without  multiplying  167  by  213. 

Write  at  sight  the  product  for  each  of  the  following  mis- 
cellaneous examples: 

31.  {x  +  5)  (x  -  5)  35.  (x  +  5)  {x  -  3a) 

32.  (x  +  5)^  36.  (x  —  3a)  {x  +  3a) 

33.  {x  —  5y  37.  (x  —  3a)  {x  +  5a) 

34.  {x  -{-  5)  (x  —  3)  38.  {x  —  5a)2 

39.  (a:  +  2/  +  5a)  (x  +  y  -  5a) 

40.  (.T  +  2/  +  5a)2     ^ 

41.  (x  +  y  +  5a)  (x  +  y  +  3a) 

42.  (x  +  y  +  5a)  (x  -\-  y  ^  3a) 

43.  (^2  + la; +  3)2 

44.  (a:2  +  i  a:  +  3)  {x'  +  ix  -  3) 

45.  {x'-i-ix  +  3)(x^  +  ix-5) 

46.  (a^  -\-  ax  -\-  x^)  (a^  —  ax  +  x^)  ^ 

47.  Make  up  and  work  an  example  in  each  principal  form 
of  abbreviated  multiplication  studied  thus  far. 

48.  How  many  of  Exs.  31-45  can  you  answer  orally? 

80.  VI.  Product  of  Two  Binomials  whose  Corresponding 
Terms  are  Similar. 

By  actual  multiplication, 
2a  -  36 
4a  +  56 


8a2  -  12a6 

+  10a6  -  156^ 
8a''  -    2ab  -  156^  Product 


120  SCHOOL  ALGEBRA 

We  see  that  the  middle  term  of  this  product  may  be  ob- 
tained directly  from  the  two  binomials  by  taking  the  alge- 
braic smn  of  the  cross  products  of  their  terms.    Thus, 

(  +  2a)  (  +  56)  +  (-  36)  (  +  4a)  =  10a6  -  12a6  =  -2a6. 

Hence,  in  general, 

The  product  of  any  two  binomials  of  the  given  form  consists 
of  three  terms: 

The  first  term  is  the  product  of  the  first  terms  of  the  binomials; 

The  third  term  is  the  product  of  the  second  terms  of  the 
binomiah; 

The  middle  term  is  formed  by  taking  the  algebraic  sum  of 
the  cross  products  of  the  terms  of  the  binomials, 

Ex.    Multiply  lOx  +  7y  by  8x  -  Uy. 

To  show  the  method  of  obtaining  the  middle  term  of  the 
product,  we  write  the  given  expression  in  the  form 


{10x+7y){Sx  -Uy) 

I I 

Hence, 

(lOx)  (-lly)  +  {7y)  {Sx)  =  -  UOxy  +  5Qxy  =  -  5^y 
.*.  (lOx  +  7y)  {Sx  -  lly)  =  80a;2  -  5^y  -  77y^  Product 

EXERCISE  30 

Write  at  sight  the  product  of  each  of  the  following  and 
check  each  result: 

1.  {2x  +  3)  (a:  +  4)  7.  {bx  -  1)  {x  +  7) 

2.  {2x  -2>){x-  4)  8.  {x  +  3z/)  (3a;  -  82/) 

3.  {2x  +  3)  (a;  -  4)  9.  (Sa^  +  b)  (4a2  -  56) 

4.  (2a;  -  3)  (a;  +  4)  10.  (a;  +  i)  (f  x  +  J) 

5.  (3a  +  5)  (2a  +  3)  11.  (a  +  .26)  (2a  -  .36) 

6.  (3a  -  5)  (2a  +  3)  12.  {\x  +  fa)  (fa;  -  |a) 


ABBREVIATED  MULTIPLICATION 


121 


13.  How  many  examples  in  Exercise  9  (p.  41)  can  you 
now  work  at  sight? 

EXERCISE  31 

Review 

Write  at  sight  the  value  of  each  of  the  following  and  check  each 
result: 


1.  (2a;  +  3)2  16. 

2.  {2x  ~  3)2  17. 

3.  (2x  +  3)  (2x  -  3)  18. 

4.  {x  +  3)  (x  -  5)  19. 

5.  (2x  +  3)  (3a;  -  5)  20. 

6.  {x  +  Sy)  {x  +  2y)  21. 

7.  (2a;  +  ZyY  22. 

8.  (2a;  +  Zy)  (3a;  -  4?/)  23. 

9.  (2a;  -  ZyY  24. 

10.  (2a;  +  3?/)  (2a;  -  Zy)  25. 

11.  (5a  -  3a;)  (4a  +  5a;)  26. 

12.  (7a;  +  32/2)2  27. 

13.  (5a2.+  3?/3)  (5a2  -  ^y^)  28. 

14.  {a?  +  3a;)  {a?  -  5a;)  29. 

15.  (2a  +  3a;  +  5)2  30. 


2a  +  3a;  +  5)  (2a  +  3a;  -  5) 
2a  +  3a;  +  5)  (2a  +  3a;  -  3) 
1  -  2a;  -  3a;2  +  x^Y 
a  +  6  +  a;  +  2/)  (a  +  6  -a;  -?/) 
a2  +  aa;  +  a;2)  (a2  -  ax  -\-  x'^) 

Aa?  +  2a  +  1)  (4a2  -  2a  +  1) 
3aa;"  -  2a"-ia;)2 
a;"  +  2a;2/"-i)2 
1  -a)2 
a  -  1)2 

-  2a;  +  32/)2 

-  2a;  -  3?/)2 
a  -  6)  ( -  a  +  6) 
a;  +  3)  ( -  a;  -  3) 

3i/)2  is  the 


31.  Why  is  it  that  the  result  of  expanding  ( —  2a; 
same  as  that  of  expanding  (2a;  +  ^yY  ? 

32.  Give  two  expressions  similar  to  those  in  Ex.  31  for  which 
the  product  is  the  same. 

33.  Why  is  (a  -  6)2  equal  to  (6  -  a)  2  ?    Make  up  two  expressions 
similar  to  these. 

34.  Make  up  and  work  an  example  in  each  principal  form  of 
abbreviated  multipUcation  studied  thus  far. 


122  SCHOOL  ALGEBRA 

'    Simplify,  using  the  methods  of  abbreviated  multipHcation  as  far 
as  possible : 

35.  (a  +  26)2  +  {a  -  26)2 

36.  (a  +  26)2  -  (a  -  26)2 

37.  (2x  -  1)2  +  (1  -  2x)2 

38.  (2x  -  1)2  -  (2x  +  1)2 

39.  (3a  -  1)2  +  (2  -  3a)  (2  +  3a) 

40.  {2x  -  7y)  {2x  +  7y)  -  4(a;  -  2yy  +  ISy  {by  -  x) 

41.  (3a;2  +  5)2  +  a;2  (10  -  3a:)  (10  +  3a;)  -  (5  +  13a;2)2 

42.  (a  -  c  +  1)  (a  +  c  -  1)  -  (o  -  1)2  +  2  (c  -  1)2 

43.  {x  +y  -  xy)  {x  -y  -  xy)  +  x'^y  -  {x  -  y^)  {x  +  y'^) 

44.  Show  that  a2  =  (a  +  6)  (a  -  6)  +  62. 

45.  By  use  of  the  relation  proved  in  Ex.  44,  obtain  the  value  of 
(7i)2  in  a  short  way. 

SuG.    We  have      (7i)2  =  (7i  +  i)  (7i  -  h)  +  {W 
=  8  X  7  +  i  =  56i  Ans. 

Using  the  method  of  Ex.  45  find  the  value  of: 

46.  (8i)2  49.    (15J)2  52.    (7.5)2  55^    (75)2 

47.  (19J)2  50.    (49i)2  53.    (19.5)2  56.    (195)2 

48.  (199i)2  51.    (99§)2  54.    (99.5)2  57.    (995)2 
'  58.   (9.7)2     (Use  (9.7)2  =  10  x  9.4  +  .32) 

59.   (9.8)2  60.    (9.6)2  g^    (4,8)2  52.   982 

63.  Find  the  value  of  (a  +  6)'  by  multiplication.  Examine  the 
result  obtained.  Make  a  rule  for  obtaining  similar  products  in  a 
short  way.    Treat  (a  —  6)^  in  the  same  way. 

64.  By  use  of  the  rule  obtained  in  Ex.  63,  write  out  by  inspection 
the  value  of  {x  +  yY- 

65.  Also  of  (a  -  xY.  66.   Of  (6  +  yY. 

Solve  the  following  equations,  using  methods  of  abbreviated 
multiplication  wherever  possible: 

67.  {2x  -  1)2  -  4a:2  =  19 

68.  {2x  +  1)2  -  {2x  -  1)2  =  16 


ABBREVIATED  DIVISION  123 

Compute  in  the  shortest  way: 

69.  The  area  of  a  field  103  rd.  long  and  97  rd.  wide. 

70.  The  area  of  a  square  field  each  side  of  which  is  98  rd. 

71.  The  cost  of  62  yd.  of  cloth  at  58^  per  yard. 

72.  The  cost  of  85  A.  of  land  at  $95  per  A. 

73.  How  many  of  the  examples  in  this  exercise  can  you  work  at 
sight? 

Abbreviated  Division 

81.  Utility  of  Abbreviated  Division.  In  certain  cases 
much  of  the  labor  of  division  may  be  saved  by  the  use  of 
mechanical  rules.  We  discover  these  rules  by  performing 
the  division  operation  in  a  typical  case,  noting  the  relation 
between  the  quantities  divided  and  the  quotient,  and  for- 
mulating this  relation  into  a  rule. 

82.  I.  Division  of  the  Difference  of  Two  Squares. 

Either  by  actual  division,  or  by  inverting  the  relation  of 
Art.  77,  p.  112,  we  obtain 

=  a  —  0         and  r-  =  a  -{-  h, 

a-\-  0  a  —  b 

Hence,  in  general  language. 

The  difference  of  the  squares  of  two  quantities  is  divisible  by 
the  sum  of  the  quantities,  arid  also  by  the  difference  of  the  quan- 
tities, the  quotients  in  the  respective  cases  being  the  difference  of 
the  quantities  and  the  sum  of  the  quantities. 

Ex.  1.    — —  =  2x  +  3y  Quotient 

2x  —  Sy 

Ex.  2.    ^^  ~  ^!^  "^  ?r  =  X-  (a  +  b)  Quotient 
x+  (a-\-  b) 

Let  the  pupil  check  the  work  in  these  examples. 


124  SCHOOL  ALGEBRA 


EXERCISE  32 

Write  at  sight  the  quotient  for  each  of  the  following,  and 
cheek  each  result: 


5. 


6. 


a  —  x 
9-4a:2 

3-2x 

x^  -  811/2 

25a:2  -  362/^ 

5a;  -  6?/2 
\%x^  -  49^ 

At?  +  7r 
25a:io  -  y^'' 

7 

a'h^  -  mM^ 

aW  +  ^(^d!" 

8. 

\^  -  V 

i^-  h 

9. 

i%a'  -  i^ 

ia2  -  i:^ 

in 

.25a2  -  .1662 

,5a-  Ah 

n 

Mx"  -  .09^2 

.2a;  4-  .3^ 

TO 

a;2  -  .256* 

5x^  -  y^  •a;  -  .562 

13.  Divide  a^  +  2a6  +  62  —  4a;2  |3y  ^  _|_  5  _  2a;  by  long 
division.  Write  the  result  of  dividing  (a  +  6)2  —  4a;2  ]^y 
a-\-h—  2x  by  the  method  of  Art.  82.  Estimate  how  much  less 
the  labor  of  the  second  process  is  than  that  of  the  first. 

14.  Make  up  and  solve  an  example  similar  to  Ex.  13. 
Obtain  in  the  shortest  way  the  quotient  for  each  of  the 

following: 

jg    (X  +  1)^  -  g'  ^    (a  -  hf  -(c-  lY 


X 

+  l  +  a 

a2- 

-  (6  -  2c)2 

a  - 

-  (6  -  2c) 

4a;4 

-  (f  +  D' 

16.   ^- -4-  19. 


17.    ^ T  n        A  20 


(a  -  6)  +  (c  -  1) 

1  -  (g  +  6  -  c)2 

1  +  (a  +  6  -  c) 

(2a  +  36)2  _  (5^  _  42^)2 


2a;2  +  (2/2  +  1)  .        {2a  +  36)  -  (5a;  -  4y) 


ABBREVIATED  DIVISION  125 

Write  a  divisor  and  quotient  for  each  of  the  following: 
^,    a2-4r'  ^^    da"  -  (x  +  yf 

21.    ^  24.  = 


a^  -  4a:«  (a  +  a;)^  -  (6  +  i/)^ 

22.    =  25.    


^  -f                                  4(a  +  6)2  -  9ir2 
23.    =  26.    


Find  two  factors  for  each  of  the  following: 

27.  2500  -  16  29.  2491  31.  99.19 

28.  2484  30.  9919  32.  6319 

33.  Divide  a^  —  6^  by  a  —  6.    Divide  (a  —  6)^  by  (a  —  6). 

34.  Find  the  difference  in  value  between  (x  +  yf  and 
aj^  +  y^i  when  a:  =  2  and  2/  =  3. 

83.  II  and  III.  Division  of  Sum  or  Difference  of  Two  Cubes. 

By  actual  division  we  can  obtain, 

0?  -j-  53  ^3  _  ^3 

— —-r-  =  a}  —  ab  -\-  6^     and  r  =  c?  ■\-  ah  ■\-  h^, 

a  +  0  a  —  0 

Hence,  in  general  language. 

The  sum  of  the  cubes  of  two  quantities  is  divisible  by  the  sum 
of  the  quantities y  and  the  quotient  is  the  square  of  the  first  quan- 
tity ^  minu^  the  'product  of  the  two  quxintities,  plus  the  square  of 
the  second  quantity;  also 

The  difference  of  the  cubes  of  two  quantities  is  divisible  by 
the  difference  of  the  quantities ^  and  the  quotient  is  the  square  of 
the  first  quantity y  plus  the  produ^ct  of  the  two  quantities,  plus  the 
square  of  the  second. 


126 

SCHOOL  ALGEBRA 

Ex.1. 

2x  -3y            2x  -  3y 

=  (2xy  +  (2x)  (32/)  +  (SyY 

=  4a^-h  Qxy  +  V  Quotient 

F,Y   91 

(«-^)'  +  27      ,„.      ,,,      ,,„,      ,,   , 

•    •      (a  -  6)  +  3        '        ^       ^        ^   ' 

=  a2-2a6  +  62-3a+36  +  9  Qwo^teni 
Let  the  pupil  check  the  work  in  these  examples. 

EXERCISE  33 

Write  at  sight  the  quotient  for  each  of  the  following  and 
check  each  result: 

a^  +  8  27a«  +  y^ 

^'  -  ^     3a2  +  y^ 

m^T?  -  f 

'2x  -  y 

h^  -  h 

11.  Divide  Sa^  +  276^  by  2a  +  36  by  the  method  of  long 
division.  Now  write  out  the  same  quotient  by  the  method 
of  Art.  83.  Estimate  how  much  of  the  labor  of  division  is 
saved  by  using  the  second  method  of  obtaining  the  quotient. 

12.  Make  up  and  work  an  example  similar  to  Ex.  11. 

13.  Treat  (a  +  hf  —  Sa:^  divided  by  (a  +  6)  —  2x  as  in 
Ex.  11. 


a+2 

x^-l 

X-  1 

27:x?  -  64 

3a: -4 

l  +  8a:« 

l  +  2ar^ 

125  -  x^ 

2. 


3.  —z 7-  a 


ABBREVIATED  DIVISION  127 

Obtain  in  the  shortest  way  the  quotient  for  each  of  the 
following: 

c3  +  (1  -  xy  (a  -  1)3  -  a^ 

14     ' — ^^ —  17     ^^ 

•    c  +  (1  -  a:)  (a  -  1)  -  x" 

8  -  (a:  +  yf  Sx^  +  jx''  -  If 

'    2-  (x-hy)  2x^x^  -\ 

21x'  +  125i/^  8(a;  -  yf  -  z' 

^^'      3r^  +  51/3  ^9-    2{x  -y)  -z 

Write  the  binomial  divisor  and  the  quotient  for 

Sa^  -  :x?                                 8a3  -fl 
20. =  24.    = 

21. =  25.    -^  = 


22.    =  26.    ^^-^  = 

23.    =  27.    ^ ^  = 

(g  +  6)3  +  (a;  +  yf  _ 
28. 

Find  a  factor  of  each  of  the  following: 

29.  203  ^  33  31;  3Q27  33,  125027 

30.  8000  +  27  32.  7973  34.  124973 

35.  Divide  c^  —  ¥  by  a  —  b.     Also  divide  (a  —  by  by 
a  —  b. 

36.  Find  the  difference  in  value  between  3^  -\-  y^  and 
(x  +  yY  when  x  =  2  and  y  =  S. 

Write  a  binomial  divisor  and  the  corresponding  quotient 
for  each  of  the  following  miscellaneous  examples: 

o,    a' -462  ^^    a3-863 

37.    =  38.    = 


128  SCHOOL  ALGEBRA 

40.  ?Z^-^  =  47.  g'  -  4(a;  +  y)'  ^ 

41.  9^-1  =  48.  g'  -  Six  +  y)»  ^ 
^2  16a^  -  9  ^  ^g  8(x  +  y)»  +  a?  ^ 
^g  8x»  -  27jf'  ^  ^  a*  -  9{x  -  yf  ^ 

44.  ^  +  ^  =  51.  27a»  -  (a;  -  yY  ^ 

45.  ^^  =  52.    (x  +  y)^  -(X-  vY  ^ 

53.   (a:  +  1)^  +  (X  -  1)^  _ 

54.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

55.  How  many  examples  in  Exercise  1  (p.  8)  can  you 
now  work  at  sight? 

84.  IV,  V,  and  VI.    Division  of  Sum  or  Difference  of  any 
Two  Like  Powers. 

By  actual  division  we  can  obtain, 


a  +  h 
a  —  b 


=  a?  —  a^b  +  ah'^  —  ¥  Quotient 
=  a^  +  a?b  +  ab'^  +  W  Quotient 


a^  +  ¥  is  not  divisible  by  either  a  +  6  or  a  —  6.    But 
«^  +  ^'  _  ^4  _  ^35  _^  ^252  _  ^53  ^  54  Quotimt 


a-\-b 


a^  +  0^5  +  a2fe2  +  a¥  +  ¥  Quotient 
a  —  0 


ABBREVIATED  DIVISION  129 

Hence, 

The  difference  of  two  like  even  powers  of  two  quantities 
is  divisible  by  the  sum  of  the  quantities,  and  also  by  their 
difference; 

The  sum  of  two  like  odd  powers  of  two  quantities  is  divisible 
by  the  sum  of  the  quantities; 

The  difference  of  two  like  odd  powers  of  two  quantities  is 
divisible  by  the  difference  of  the  quantities. 

For  the  quotient  in  all  these  cases  — 

(1)  The  number  of  terms  in  a  quotient  equals  the  degree 
of  the  powers  whose  sum  or  difference  is  divided; 

(2)  The  terms  of  each  quotient  are  homogeneous  (since 
the  exponent  of  a  decreases  by  1  in  each  term,  and  that 
of  b  increases  by  1  in  each  term). 

(3)  If  the  divisor  is  a  difference,  the  signs  of  the  quotient  are 
all  plus;  if  the  divisor  is  a  sum,  the  signs  of  the  quotient  are 
alternately  plu^  and  rrmms. 

In  the  above  statements  the  parts  in  italics  should  be 
committed  to  memory. 

The  last  statement  forms  a  general  rule  for  signs  of  a 
quotient  in  all  the  cases  of  abbreviated  division,  including 
I-VI. 

g^  ^   320:^  +  2/^  ^  {2xY  +  l^ 

'    '    2x  +  y  2x  +  y 

-  (2xy  -  {2xfy  +  {2x)Y  -  {2x)f  +  t/* 

=  16a:4-8a;3|/  +  AxY  -  2xi^  +  y^  Quotient 

a^o  +  x^^  {a'f  +  {x'f 


Ex.2. 


a^  -\-  x^  a^  -\-  x^ 


130  SCHOOL  ALGEBRA 


EXERCISE  34 


Write  at  sight  the  quotient  for  each  of  the  following  and 
check  each  result: 


1. 

a^-ha:^ 

a  -}-  X 

2. 

a^-3^ 

a  —  X 

3. 

b'-\-y' 

h  +  y 

a. 

h'  -  y' 

c 

a5  +  32 

a  +  2 

6. 

a?  -  128 

a-  2 

7. 

7^  -  1 

X-  1 

o 

0:^  +  1 

9. 


10. 


11. 


12. 


Z2x> 

-f 

2x- 

-y 

a"  +  ar^i 

a  + 

X 

x^^- 

ylS 

x"-- 

f 

243  - 

-d^ 

b  -  y  X  -^  I 

13.  Divide  32a^  -]-  x^  hy  2a  -\-  x  hy  the  method  of  long 
division.  Now  write  the  same  quotient  by  the  method  of 
Art.  84,  p.  128.  Estimate  how  much  of  the  labor  of  division  is 
saved  by  using  the  second  method. 

14.  Make  up  and  work  an  example  similar  to  Ex.  13. 
Write  a  binomial  divisor  and  the  corresponding  quotient 

for  each  of  the  following: 
_    a'-^-y"  _  ^^    a^-32_ 


16. 
17. 
-io 

o'- 

y" 

fc'  + 

x' 

6'- 

x- 

20. 
21. 


a?  +  128 


1  -  32a:5 


2/^  +  1 


22. 

Obtain  a  factor  of  each  of  the  following: 
23.   100,001  25.   100,032 

6fk.  100,243  26.  99,757 


ABBREVIATED  DIVISION  131 

27.  Divide  a^  +  U"  by  a  +  &.     Also  divide  (a  +  hY  by 
a +  6. 

28.  Find  the  difference  in  value  between  ^  —  -if"  and 
{x  —  y)^,  when  x  =  3  and  y  —  2, 

EXERCISE  36 

Review 
Write  at  sight  the  quotient  for  each  of  the  following: 


1 

62  _a:2 

6  -X 

o 

¥  -x^ 

b-x 

•5 

¥  -x^ 

b   -X 

4 

¥  +x^ 

b+x 

(% 

¥  +x^ 

6  H-2x 


11. 


7     ^'  -  ^^^  12 

b  -2x 


¥  -  32a;5 
b  -2x 


13. 


x'  - 

8(a 

+  by 

X  - 

2(a 

+  b) 

Sx'  ■ 

-a3 

2x 

-  a 

27a« 

-8{ 

[x  +  yY 

3a2 

-  2{x  +  y) 

X''  +  y' 

x'-\-y^ 

a;i2- 

.yS 

9.    ^^+^^^  14. 

b  +2x 

10  ^'  -  4(«  +  by    ^g 

"'6+0;  ■    a:  -  2(a  +  &)  '    x^  +  y^ 

16.  In  Ex.  11  remove  the  parenthesis  in  the  dividend  and  divisor, 
and  divide  by  long  division.  The  work  required  is  about  how 
many  times  that  required  in  the  abbreviated  process? 

Write  a  binomial  divisor  and  the  corresponding  quotient  for  each 
of  the  following: 

j^j    b'  -  82/3  ^  ^^    a?  -  4(x  +  yY 

18.   ^1^^  =  25. 


19    ¥+%y^  _ 


26. 


20.  ^^-^^  =  27. 

21.  §?1±27  ^  28. 

22.  ^^^  =  29. 
23.^^^=^  30. 


a3  +  8(a: 

+  yY 

a?  -  8(x 

+  yy 

8x«+2/« 

4x*  -'f 

32a«  -  t/ 

5 

8X6    _y^ 

132  SCHOOL  ALGEBRA 

Divide  each  of  the  following  by  a;  -  o  in  a  short  way: 

31.  a:3  -  a3  +  x2  -  a^  34.   3(a;3  -  a^)  +  4(a;  -  a) 

32.  x^  -  a?  +  5(a;2  -  a^)  35.   (a;  -  af  +  5(^2  -  a^) 

33.  x'  -  a'  +  5(x  -  a)  36.   7(a;  -  a)^  +  b{x  -  a) 
Find  the  value  of  each  of  the  following  in  the  shortest  way: 

37.  (o  +  6)  (a  +  6)  (a  -  6)  (a  -  6) 

38.  (a  +  26)  (a  +  26)  (a  -  26)  (a  -  26) 

39.  (3x  -  22/)  (3a;  -  2y)  (3a;  +  2y)  (3a;  +  22/) 
Simphfy: 

40.  bx  -  Z{x  -  2)2  -  3(3  -  2a;)  (1  +  x) 

41.  7  -  5(a;  -  2Y  -  3(3  -  2a;)  {- x) 
Solve: 

42.  (a;  -  8)  (a;  +  12)  _  (x  +  1)  (a;  -  6)  =  0 

43.  (2a;  -  1)  (a;  +  3)  -  (a;  -  3)  (2a;  -  3)  =  72 

46.  Four  times  a  certain  number  diminished  by  12.07,  equals 
twice  the  number  increased  by  1.13.    Find  the  number. 

47.  Separate  1000  into  three  parts  such  that  the  second  part  is 
three  times  as  large  as  the  first  part,  and  the  third  part  exceeds  the 
first  part  by  100. 

f^  48.  The  Suez  Canal  is  100  miles  long.  This  is  2  miles  more  than 
8  times  the  length  of  the  Simplon  Tunnel.  Find  the  length  of  the 
tunnel. 

49.  The  temperature  of  the  electric  arc  is  5400°  F.  This  is  464° 
more  than  8  times  the  temperature  at  which  lead  fuses.  Find  the 
temperature  at  which  lead  fuses. 

50.  The  velocity  of  sound  in  the  air  is  1090  ft.  per  second.  This 
rate  is  10  ft.  more  than  9  times  the  rate  at  which  sensation  travels 
along  a  nerve.  Find  the  rate  at  which  sensation  travels.  How 
does  this  compare  with  the  velocity  of  an  express  train  going  60 
miles  per  hour? 

51.  Who  first  used  the  sign  +  to  denote  addition,  and  when? 
(Seep.  268.) 


ABBREVIATED  DIVISION  133 

52.  Give  some  other  symbols  used  to  represent  addition  before 
fche  sign  +  was  invented.  Discuss  as  far  as  you  can  the  relative 
advantages  of  these  signs. 

53.  Answer  the  questions  in  Ex.  52  for  the  subtraction  sign. 

54.  Answer  the  questions  in  Ex.  52  for  the  sign  X. 

55.  Answer  the  questions  in  Ex.  52  for  the  sign  t-. 

56.  Answer  the  questions  in  Ex.  52  for  the  sign  =. 


CHAPTER   VIII 

FACTORING 

85.  The  Factors  of  an  expression  (see  Art.  11)  are  the 
quantities  which,  multipHed  together,  produce  the  given 
expression. 

Factoring  is  the  process  of  separating  an  algebraic  expres- 
sion into  its  factors. 


86.  Utility  of  Factoring     If  it  is  known  that 

V  -    8a:  +  15  =  (  a:  -  3)  (  a:  -  5) 

and          2x2  _  13^  +  21  =  (2a:  -  7)  (  a:  -  3) 

a:^  -    8a:  +  15       (  a:  -  3)  (  a:  -  5)         x- 
^"^       2a:2  _  13^.  _^  21       {x-  3)  (2a:  -  7)       2a:  - 

-5 

-7 

The  above  reduction  of  a  fraction  to  a  simpler  form  illus- 
trates the  usefulness  of  a  knowledge  of  factoring  in  enabling 
us  to  simplify  work  and  save  labor. 

Why  do  we  now  proceed  to  make  definitions  and  rules  and 
to  divide  the  topic,  Factoring,  into  cases? 

87.  A  Prime  Quantity  in  algebra  is  one  which  cannot  be 
^divided  l)y  any  quantity  except  itself  and  unity;  as  a,  b, 

a"  +  b^  17. 

In  all  work  in  factoring,  prime  factors  are  sought,  unless 
the  contrary  is  stated. 

88.  Perfect  Square  and  Perfect  Cube.  When  an  expres- 
sion is  separable  into  two  equal  factors,  the  expression  is 

134 


TTACTOHnSIG  135 

called  a  'perfect  square,  and  each  of  the  factors  is  the  square 
root  of  the  expression. 

Thus,  9aV  =  3ax2  •  Sax*. 

.*.  3aa;2  jg  j;he  square  root  of  Oa^x*. 

Also,  x2  -  4x  +  4  =  (x  -  2)  (x  -  2),  and  is  therefore  a  perfect 
square,  with  a;  —  2  for  its  square  root. 

When  an  expression  is  separable  into  three  equal  factors, 
the  expression  is  called  a  perfect  cube,  and  each  of  the  factors 
is  its  cube  root. 

Thus,  27a^xY  =  ^(^^Y '  ^(^^Y '  ^a^y. 

.'.  Saxhj^  is  the  cube  root  of  27 a^x^y^. 

89.  The  Factors  of  a  Monomial  are  recognized  by  direct 
inspection. 

Thus,  the  factors  of  7aV  are  7,  a,  a,  x,  x,  x. 

90.  Factors  of  Polynomials.  Multiply  ix^  +  2xy  +  y^  fcy 
4:x'^  —  2xy  +  2/^.  What  terms  are  canceled  in  adding  the 
partial  products?  Because  these  terms  have  been  thus  can- 
celed and  have  disappeared,  it  is  difficult  to  take  the  final 
product  16a;'*  +  A.x^y'^  +  y^  and  from  it  discover  the  original 
factors  which  were  multiplied  together  to  produce  it. 

Hence,  in  factoring  polynomials  various  methods  must  be 
devised  to  meet  different  cases,  and  the  cases  must  be  care- 
fully discriminated. 

Case  I 

91.  A  Polynomial  having  a  Common  Factor  in  all  its  Terms. 
Ex.    Factor  3x2  ^  e^.^ 

3x2  4-  6a;  =  Zx{x  +  2)  Factors 

At  first,  in  working  examples  of  this  kind,  it  is  well  to  put  the 
work  in  the  following  form: 

3a;)3a;2  +  6a;  =  3x(a;  +  2)  Factors 
X  +2 


136  SCHOOL  ALGEBRA 

Check  by  Substitution      If  we  let  a;  =    2 

Sx{x  +  2)  =  6(2  +  2)  =24 

Also  3^2  +  6a;  =  12  +  12  =  24 

Check  by  Multiplication 

X  +2 
Sx 


3^2  +  6a; 
Hence,  in  general, 

Divide  all  the  terms  of  the  polynomial  by  their  largest  common 
factor; 

The  factors  will  be  the  divisor  and  quotient, 

EXERCISE  36 

Factor  the  following  and  check  the  work  for  each  example 
cither  by  substitution  or  by  multiplication,  or  by  both  as 
the  teacher  may  direct: 


1. 

2:x^  +  50^2 

6. 

Sa^x^  -  15a'^s^ 

11.  ia'b  -  f  a62 

2. 

;i^-2x 

7. 

ISa:^  -  27x'y 

12.    U^  -  2x^ 

3. 

x^-^x 

8. 

x'-7?-x' 

13.   .2:x?  +  Aax^ 

4. 

Sa^-a 

9. 

aH  -  2aV 

14.   max^  -  AaV 

5. 

7a  +  Ua^ 

10. 

h:^  +  \^ 

15.   1.2mn  —  .6^2 

16. 

3a2  -  Qax  +  9^2 

19.  a^bh- 

-  a^tV  +  2a26^c2 

17. 

2x  +  4^2  - 

63^ 

20.  2a;V 

-  Sx^Y  +  60:3 Y 

18. 

lOa^b^  -  S5aW 

21.  a^6V 

"  +  lla-6V»+i 

22.  7(a  +  b)x  +  5(a  +  b)y 

23.  7{a-\-b)x^y-\-b{a-{-byxy'^ 

24.  21  (a;  -  yf  -  U{x  -  yf 

25.  9(2a;  -  a)  3-  12(2a;  -  af 


FACTORING  137 


26.  847  X915  -  847  X  913 

t27.  312.75  X  87  -  312.75  X  84 
28.  8  X  11  X  232  +  7  X  11  X  232  -  5  X  11  X  232 

29.  7rE2  -j-  7rr2  when  ir  =  -\2-,  R  =  8,  and  r  =  6. 

30.  -n-R^  -  7rr2  when  tt  =  ^j\  R  =  410,  and  r  =  60. 
Find  the  value  of  x  in  the  following  equations: 

31.  ax  -\-  bx  =  10.  (What  does  the  value  of  x  become 
when  a  =  5  and  b  =  15?) 

32.  ax  =  10  —  6a: 

33.  ax  -\-  bx  -\-  ex  =  12 

34.  2ax  —  bx  +  3cx  =  15 

Factor  the  numerator  and  denominator  of  each  of  the  fol- 
lowing fractions  and  then  simplify  the  fraction  by  canceling 
factors : 

Sa^b  —  6ab'^  aV 

35.  — 37. 


3a26  +  6a62  4x^  -  Qa^ 

3g     x^  +  2x'    ^  gg      Spq  -  6pV 


3x^  -  6x2  12pY  _  Qpq 

39.  From  an  examination  of  Exs.  26-38,  state  the  uses  or 
advantages  of  being  able  to  factor  by  the  method  of  Case  I. 

40.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

Case  II 

92.  A  Trinomial  that  is  a  Perfect  Square.  By  Arts.  75 
and  76  a  trinomial  is  a  perfect  square  when  its  jBrst  and  last 
terms  are  perfect  squares  and  positive,  and  the  middle  term  is 
twice  the  product  of  the  square  roots  of  the  end  terms.    The 


138  SCHOOL  ALGEBRA 

sign  of  the  middle  term  determines  whether  the  square  root 
of  the  trinomial  is  a  sum  or  a  difference. 

Ex.  1.    Factor  IQx^  -  24xy  +  V. 

16x2  -  24xy  +  9?/2  =  (4a;  -  Sy)  {4x  -  Sy)  Factors 

Ex.  2.    Factor  (a  +  by  +  4(a  +  b)x  +  4x2. 

(a  +  by  +  4(a  +  fe)x  +  4x^  =  [{a  +b)  +  2a:p 

=  (a  +  6  +  2xy  Ans. 

Hence,  in  general,  to  factor  a  trinomial  that  is  a  perfect 
square. 

Take  the  square  roots  of  the  first  and  last  terms,  and  connect 
these  by  the  sign  of  the  middle  term; 

Take  the  result  as  a  factor  twice, 

EXERCISE  37 
Factor  and  check: 

1.  4^^  +  4x2/ +  2/2  9.  a5  +  2a^4-a' 

2.  16o2  -  2^ay  +  V  10.  ^3?  +  44a:y  4-  i2lxy^ 

3.  25a:2  -  lOx  +  1  11.  SWb  +  \2^a^b''  +  4Qa}^ 

4.  x^  -  20xy  +  100y2  12.  Sa^y  -  40axy  +  50x^y 

5.  49c+286c2  +  46V  13.  2x^  -  Sx^  +  Sx^ 

6.  a362  +  4a36  +  4a3  14.  30x2?/ +  3x^  +  75y2 

7.  x?/2  +  2x1/  +  X  15.  a^x  +  ax^  —  2a2x2 

8.  2m'^n  —  4m?i  +  2n  16.  x^"  +  2x"!/  +  y^ 

17.  (a  -  by  -  2c{a  -  6)  +  c^ 

18.  9(x  +  yy  +  12z(x  +  2/)  +  4^2 

19.  16(2a  -  3)2  -  16a6  +  246  +  6^ 

20.  25(x  -  I/)'  -  120x!/(x  -  y)  +  144x2/ 

21.  a2  +  62  +  c2  +  2a6  +  2ac  +  26c 


FACTORING  139 

22.  i'r  +  4xy-\-  V  24.  .04a2  -  .12a6  +  .0962 

23.  iir2  +  ixy  +  jy  25.  25a2  -  30ax  +  90:^ 
Solve  the  following  equations  for  x: 

26.  ax  -  Sbx  =  a^  -  Qab  +  96^ 

27.  aa:  +  36a;  =  a^  +  6a6  +  96^ 

28.  X  —  2ax  =  1  —  4:ax  -  4a2 

29.  2ax  +  36a;  =  4a2  +  12a6  +  96^ 

Factor  the  following  examples  in  Cases  I  and  II,  and  check 
each  result: 

30.  a;3  _  4^.2  ^  4^.  36    iQa^  -  20a  +  10 

31.  x^  -3x^  +  4x  37.  20a2  -  10a  +  10 

32.  m^  —  2m^n  +  'm^n^  38.   IGa^p^  _  24a^p  +  90^ 
v33.  m^  —  m%  +  m^n^  39.   4a;^  +  4:3^y  +  a;^?/^ 

34.  a%2  _  8^3^  _^  15^3  4Q    83^  _^  163.22^  ^  ^2/4 

35.  o''^a;2  —  Qa^x  +  4a2  41.    IGm^Ti^  —  9mn^  +  n^ 

42.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

Case  III 
93.  The  Difference  of  Two  Perfect  Squares. 

From  Art.  77,  p.  112,     (a  +  6)  (a  -  6)  =  a^  -  6^ 
Hence,  a^  —  6^  =  (a  +  6)  (a  —  6) 

But  any  algebraic  quantities  may  be  used  instead  of  a  and 
6.     Hence,  in  general,  to  factor  the  difference  of  two  squares, 
Take  the  square  root  of  each  square; 
The  factors  will  he  the  sum  of  these  roots  and  their  difference. 


140  SCHOOL  ALGEBRA 

Ex.  1.    Factor  x^  -  16?/2. 

x^  -  162/2  =  (a:  +  %)  {x  -  Ay)  Factors 

Ex.  2.    Factor  x^  -  y\ 

X*  -y*  =  (x^  +  2/2)  (x2  -  2/2) 

=  (a;2  +  2/2)  (x  +  y)  (x  -  y)  Factors 

Ex.  3.    Factor  (3a:  +  Ayf  -  {2x  +  ?^yf. 

(3x  +  42/)2  -  (2a;  +  ZyY  =  [(3a:  +  Ay)  +  (2x  +  32/)]  [(3x  +  42/)  - 

(2a:  +  32/)] 
=  (3a:  +  42/  +  2a:  +  32/)  (3a:  +  42/  -2a;  -Zy) 
=  (5a:  +  72/)  {x  +  2/)  Factors 
Let  the  pupil  check  the  above  examples. 

EXERCISE  38 
Factor  and  check: 

1.  a:2  -  9  10.  x^  -  ^arx?  19.  a^^  -  a^ 

2.  25  -  16a2       11.'  m  -  64a27/i  20.  a^x  -  x 

3.  4a2  -  4962      12.  242  -  2x'^  21.  225a:2"  -  2/2 

4.  a:^  -  42/2  13.  x^  -  X?  22.  2ia:2  -  ^2/2 

5.  100  -  81m2    14.  2.3?  -  75xy^  23.  2*5^2  -  962 

24.  .09a:2  _   iQy2 

25.  .01a2  -  .0462 

26.  .252/2-^^2^2 

27.  .81a:2  _  .002562 
35.    (a:  +  22/)2  -  (3a:  +  1)2 

29.  a:^"  -  2/2"26  36.  25(2a  -  6)2  -  (a  -  36)2 

30.  (a:  +  2/)2  -  1  37.  a:i22/9  -  2/2^6 

31.  a:2  -  (2/  +  1)2  38.  81a:i2  _  iQyA 

32.  (a:  -  2/)^  -  9  39.  x^  -  lAAxyh^ 

33.  4(a:  -  2/)2  -  25  40.  (a  -  6)2  -  4(c  +  1)2 

34.  1  -  36(a:  +  2y)^  41.  1  -  100(a:2  -  a:  -  1)2 


6. 

9a'  -  x' 

15. 

a'-x' 

7. 

1  -  64m2 

16. 

a^  -  816^ 

8. 

3a:2  _  122/2 

17. 

^8     _     y% 

9. 

X?  -  Wx 

18. 

x'-x 

28. 

x'-f 

35. 

II  FACTORING  141 

Solve  the  following  equations  for  x: 

42.  ax  +  26a;  =  a^  —  46^      44.  ?>x  —  ax  =  ^  —  a^ 

43.  ax  —  2bx  =  a^  —  46^        45.  x  —  bx  =  1  —  b^ 
Factor  the  following  miscellaneous  examples: 

46.  a^  —  4:a  53.  a^  —  9x 

47.  a^  —  4  .  54.  a^  +  9a^x  +  6ax^ 

48.  a^  -  4a  +  4  55.  a^  +  Qa^x  +  Qax^ 

49.  a^  —  4a  56.   a^  —  ax'^ 

50.  a'  —  4a2  +  4a  57.  a^  +  ax* 

51.  a*  -  4a3  +  4a2  58.    (a  +  x)^  -  9 

52.  0:^-60:  59.   (a  +  6)2  -  (x  -  2/  -  a)2 

60.  Make  up  and  work  an  example  in  factoring  in  each  of 
the  cases  treated  thus  far. 

61.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

62.  How  many  of  the  examples  in  Exercise  2  (p.  13)  can 
you  now  work  at  sight? 

Case  IV 

94.   A  Trinominal  of  the  Form  oc^  +  bx-\-c. 

It  was  found  in  Art.  79  (p.  117)  that  on  multiplying  two 
binomials  like  x  +  3  and  x  —  5,  the  product,  x"^  —  2x  —  15, 
was  formed  by  taking  the  algebraic  sum  of  +  3  and  —  5  for 
the  coefficient  of  x  (viz.  —  2),  and  taking  their  product  (—15) 
for  the  last  term  of  the  result.  Hence,  in  undoing  this  work 
to  find  the  factors  of  x^  —  2x  —  15,  the  essential  part  of  the 
process  is  to  find  two  numbers  which,  added  together,  will 
give  —  2  and,  multiplied  together,  will  give  —  15. 


142  SCHOOL  ALGEBRA 

Ex.  1.    Factor  a:^  +  n^  +  30. 

The  pairs  of  numbers  whose  product  is  30  are  30  and  1,  15  and  2, 
10  and  3,  6  and  5.    Of  these,  that  pair  whose  sum  is  11  is  6  and  5. 

Hence,  x^  +  llx  +  SO  =  (x  +  Q)  {x  -{-  5)  Factors 

Ex.  2.    Factor  x^  -  x  -  30. 

It  is  necessary  to  find  two  numbers  whose  product  is  —  30,  and 
whose  sum  is  —  1. 

Since  the  sign  of  the  last  term  is  minus,  the  two  numbers  must 
be  one  positive  and  the  other  negative;  and  since  their  sum  is  —  1 
the  greater  number  must  be  negative. 

x^  -X  -SO  =  {x  -Q)  (x  +6)  Factors 

Ex.  3.    Factor  x^  +  Sxy  -  10/. 

Since  5y  and  —  2y,  added  give  Sy,  and  multipUed  give  —  lOy^, 
x2  +  Sxy  -  10?/2  =  (a;  +  5y)  {x  -  2y)  Factors 

Hence,  in  general,  to  factor  a  trinomial  of  the  form 

7?    -{•     hX     ■}-     Cy 

Find  two  numbers  which,  multiplied  together,  produce  the 
third  term  of  the  trinomial  and,  added  together,  give  the  coefficient 
of  the  second  term; 

X  {or  whatever  takes  the  place  of  x)  plu^  the  one  number,  and 
X  plus  the  other  number,  are  the  factors  required, 

EXERCISE  39 

Factor  and  check: 

1.  x^-{-5x-\-6  e.  x^  +  x-SO 

2.  x^  —  X  —  6  7.  a;^  +  6xy  —  16/ 

3.  7?  -^  X  —  ^  B.  x^  —  6xy  —  16/ 

4.  x^  +  7x-44:  9.  x^  +  Sx-i-  16 

5.  x^  -  llx  +  30  10.  x^  +  5x-  3fi 


I 


FACTORING  143 

x''-5x-  36  19.  xY  -  23xy  +  132 

a;^  -  5a;2  -  36  20.  x^  -  5ax  -  2Aa^ 


13.  x^  +  3x-28  21.  a;^  -  9a:2  +  8 

14.  a:2  -  2a;  -  48  22.  2a  -  Uax  -  QOax^ 

15.  a:2  -  8a:  -  48  23.  20:^  _  22a:2  -  120a: 

16.  x^  +  13a:  -  48  24.  a:^  -  25a:3  +  144a: 

17.  x^  -  22a:  -  48  25.  a:^"  -  a:"  -  56 

18.  a:2  -  4a:  -  96  26.  aW  -  llabc^  -  26c^ 

27.  x^  -\-  (a  +  h)x  +  a6 

28.  a:2  +  (2a  -  36)a:  -  6a6 

29.  a:2  +  (a  +  26  +  c)x  +  (a  +  6)  (6  +  c) 

30.  a:2  +  (a  +  h)x  +  (a  -  c)  (6  +  c) 

31.  (a:  -  yY  -  3(a:  -  2/)  -  18 

Factor  and   check  each  of  the  following  miscellaneous 
examples : 

32.  a:2  -  4a:  +  4  38.  a"^  -  4y^ 

33.  a:^  —  4  39.  a^  —  A^a^y  +  a^ 

34.  a:^  _  4a;  -I-  3  4q  a^  -  1 

35.  a:^  —  a:^  —  6a:  41.  x^  +  bax  +  6a2 

36.  a:^  —  4a:  42.  a:  —  a:^ 

37.  a:^  +  6a:2  +  g^  43  ^4  _  7^2  ^  12 

44.  Make  up  and  work  an  example  in  factoring  to  illus- 
trate each  case  treated  thus  far. 

45.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 


144  SCHOOL  ALGEBRA 

Case  V 

95.   A  Trinomial  of  the  Form  ax^  -\-hx-\-  c. 

From  Art.  80  (p.  119)  it  is  evident  that  the  essential  part 
of  the  process  of  factoring  a  trinomial  of  the  form  ax^  -\-hx-\-  c 
lies  in  determining  two  factors  of  the  first  term  and  two  factors 
of  the  last  term,  such  that  the  algebraic  sum  of  the  cross 
products  of  these  factors  equals  the  middle  term  of  the 
trinomial. 

Ex.    Factor  lOa:^  +  13a;  -  3. 

The  possible  factors  of  the  first  term  are  10a;  and  x,  bx  and  2x. 
The  possible  factors  of  the  third  term  are  —  3  and  1,  3  and  —  1.  In 
order  to  determine  which  of  these  pairs  will  give  +  13a:  as  the  sum 
of  their  cross  products,  it  is  convenient  to  arrange  the  pairs  thus: 

lOx,  -  3  bx,  -I 

X,       1  2x,       3 

Variations  md,y  be  made  mentally  by  transferring  the  minus  sign 
from  3  to  1;  and  also  by  interchanging  the  3  and  the  1. 
It  is  found  that  the  sum  of  the  cross  products  of 

5x,  -  1       . 

IS  +  13a; 
2a;,      3 

Hence,        lOx^  +  13a;  -  3  =  (5a;  -  1)  {2x  +  3)  Fact(yrs 
Let  the  pupil  check  the  work. 

Hence,  in  general,  to  factor  a  trinomial  of  the  form 

ax^  -\-hz  -\-  Cy 

Separate  the  first  term  into  two  siich  factors,  and  the  third 
term  into  two  such  factors,  that  the  sum  of  their  cross  products 
equals  the  middle  term  of  the  trinomial; 

As  arranged  for  cross  multiplication,  the  upper  pair  taken 
together  and  the  lower  pair  taken  together  form  the  two  factors. 


I 


FACTORING  146 

EXERCISE  40 


Factor  and  check: 

1.  2x2  4-  3a;  -f  1  15.  6a;2y  -  2xy  -  4y 

2.  3a:2  -  14x  +  8  16.   IGx^  -  60:2/  -  21y^ 

3.  2a:2  +  5a;  +  2  17.   12a:2  +  a;^  _  63?/2 

4.  3a:2  +  lOx  +  3  18.   32a2  +  4a6  -  456^ 

5.  6a:2  -  7a;  -  5  19,  4^.4  _  13^,2  _^  9 

6.  2a;2  +  5a;  -  3  20.  9a;^  -  148a;2  +  64 

7.  6a;3  +  20a;2  -  16a;  21.   12a;2  -  7xz  -  12z^ 

8.  3a;^  -ij^-4x^  22.   24a;3  _|_  i04a;V  -  18a;?/^ 

9.  8a2  +  2a  -  15  23.  25a4  +  9a^b^  -  16¥ 

10.  2a;2  +  a;  -  10  24.    16a;^  -  lOa:^?/^  -  9y^ 

11.  12a;2  -  5a;  -  2  25.  3a;2"  -  8a;"i/  -  3y^ 

12.  4x2  +  11a;  -  3  26.   25a4  -  41^252  +  166* 

13.  5a;2  +  24x  -  5  27.  20  -  9a;  -  20a;2 

14.  93^  -  15a;2  -  6a;     '         28.   5  +  ^2xy  -  2\xY 

29.  (a  +  6)2  +  5(a  +  6)  -24 

30.  3(a;  -  yy-  +  l{x  -  y)z  -  ^z^ 

31.  3(a;2  +  2a;)2  -  5(^2  +  2a;)  -  12 

32.  4a;(a;2  +  3a;)2  -  8a;(x2  +  3a;)  -  32a; 

33.  2(a;  +  1)2  -  5(a;2  -  1)  -  3(a;  -  1)2 

Factor  and  check  each  of  the  following  miscellaneous 
examples: 

34.  4a;2  -  1  38.  a;'*  -  1 

35.  4ar^  +  4a;  +  1  39.  a;^  —  a;2  —  6a; 

36.  3a;2  +  4a;  +  1  40.  5a2  +  9a  -  2 

37.  a;2  +  4a;  +  3  41.  or^  -  9a;  +  18 


46. 

x'-a^ 

47. 

x'  -X 

48. 

x^  -  ax  -  2a^ 

49. 

27?  -  bx^  -  Zx 

146  SCHOOL  ALGEBRA 

42.  a:^  _  6^2  +  9a; 

43.  a2  -  4(a:  +  yf 

44.  3a:2  +  7a;  _  6 

45.  (a  +  &)2  +  2(a  +  6)a:  +  a:2 

50.  Make  up  and  work  an  example  in  factoring  in  each 
case  treated  thus  far. 

51.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

52.  How  many  examples  in  Exercise  25  (p.  101)  can  you 
work  at  sight? 

Case  VI 
96.   Sum  or  Difference  of  Two  Cubes. 

From  Art.  83  (p.  125),  ^5!±_^  ^  a"  -  ah -\-  h\ 

a  -jr  0 

Hence,  a^ -^  ¥  =  (a  +  b)  {a^  -  ab  +  ¥)  ,  .  .  (1) 

In  like  manner,  a^  -  ¥  =  {a  -  b)  (a^  -^  ab  +  b^)  .  .  .  (2) 

But  any  algebraic  expressions  may  be  used  instead  of  a  and 
b  in  formulas  (1)  and  (2). 

Ex.  1.    Factor  27a:3  -  Sf, 

27x3  -8z/3  =  (Sxy  -  {2yY 

Use  3x  for  a  and  2y  for  h  in  (2)  above. 

27a;3  -  8i/3  =  (3a;  -  2y)  {^x^  +  ^xy  +  4^/^)  Factors 

In  working  examples  of  this  type,  it  is  often  convenient  to  call 
3a;  —  2y  the  "divisor  factor"  and  Oa;^  +  ^xy  +  4?/^  the  "quotient 
factor."    Why  are  these  names  appropriate  in  this  case? 

Ex.  2.    Factor  a«  +  86^ 

a«  +  869  =  (^2)3  _|_  (263)3 

=  (a2  +  263)  (a*  -  20^6^  +  46^)  Factors 


I 


FACTORING  147 


Ex.  3.    Factor  (a  +  by  -  s^. 

(a+by  -3^  =  [{a  +  b)  -  x]  [{a  +  by  +  (a  +  b)x  +  x"] 
Let  the  pupil  check  the  above  examples. 

Hence,  in  general,  to  factor  the  sum  or  difference  of  two 
cubes. 

Obtain  the  values  of  a  and  b  in  the  given  example,  and  substi-- 
tute  these  values  in  formula  (1)  or  (2). 

97.  Sum  or  Difference  of  Two  Like  Odd  Powers. 

Since  the  difference  of  two  like  odd  powers  is  alwp.ys  divis- 
ible by  the  difference  of  their  roots  (see  Art.  84,  p.  128),  the 
factors  of  a"  —  6",  when  n  is  odd,  are  the  divisor,  a  —  b,  and 
the  quotient. 

Ex.  1.    Factor  a^  -  ¥. 

a5  -  6^  =  (a  -  b)  (a*  +  a%  +  a%^  +  a¥  +  ¥) 

Since  the  sum  of  two  like  odd  powers  is  divisible  by  the 
sum  of  the  roots  (see  Art.  84,  p.  128),  the  factors  of  a"  +  &", 
when  n  is  odd,  are  the  divisor,  a  -{-  b,  and  the  quotient. 

Ex.  2.    Factor  a^  +  32yK 

x^  +  32?/5  =x'  +  {2yy 

=  (x  +  2y)  [x*  -  xK2y)  +  x\2yy  -  x{2yy  +  {2yy] 
=  {x  +  2y)  {t^  -  2x^y  +  ^"^y"^  -  8xy^  +  IQy*)  Factors 

98.  Sum  or  Difference  of  Two  Like  Even  Powers. 

The  difference  of  two  like  even  powers  is  factored  to  best 
advantage  by  Case  III  (p.  139). 

Ex.  1.    x^  —  y^, 

=  {x'  +  y')  (x'  +  y')  (x  -\-y){x-  y)  Factors 

The  sum  of  two  like  even  powers  cannot  in  general  be  fac- 
tored by  elementary  methods  unless  the  expression  may  be 


148  SCHOOL  ALGEBRA 

regarded  as  the  sum  or  difference  of  two  cubes  (Art.  96),  or 
other  Uke  odd  powers. 

Ex.  2.   a«  +  66  =  {a^f  +  Qy'f 

=  (a2  +  62)  (a^  -  a^h^  +  6^)  Factors 

But  a^  +  6^,  a*  +  6^  and  a^  -\-  ¥  cannot  be  factored  by  any 
elementary  method,  and  are  therefore  prime  expressions. 

Let  the  pupil  check  the  examples  of  Art.  97  and  98. 


I* 

I 


EXERCISE  41 

Factor  and  check: 

1.  m'  —  n^ 

14. 

a«  -  64n^ 

27. 

a"  +  x^i 

2.  c^  +  8d^ 

15. 

250a:  -  2a:7 

28. 

a«  +  69 

Z.TI  -7? 

16. 

8a:«  +  2/3 

29. 

32a:5_  i 

4.  a^  +  86^03 

17. 

(a  +  6)3  +  1 

30. 

a^i  -  611 

5.  a:^  -  125 

18. 

125  +  (26  -  af 

31. 

243 -a:^ 

6.  64!/3  _  27 

19. 

8  -  (c  +  d[)3 

32. 

64  -  (a  -  6)3 

7.  aW  +  1 

20. 

{x  -  yf  -  21^ 

33. 

8(0:  -  2yf  +  1 

8.  1  -  lOOOa:^ 

21. 

16a:y  -  54a:z3 

34. 

^10  _  510 

9.  27a;^  +  a^x 

22. 

x^  +  2/^ 

35. 

a}^  +  610 

10.  512x3  -  y^ 

23. 

0:^-2/' 

36. 

32a:5  _  ^10 

11.  a  +  343a4 

24. 

a«  +  m« 

37. 

a«  +  ^9 

12.  a^  —  x^ 

25. 

a:i2  +  2/i2 

38. 

8.t:12  +  2/9 

13.  a;i2  _  y^ 

26. 

a^  -  1286^ 

39. 

512a:3  _  (a  +  6) 

40.  Make  up  a  binomial  expression  whose  terms  contain 
unlike  exponents  and  which  can  be  factored  as  the  sum  of  two 
cubes.  Also  one  that  can  be  factored  as  the  sum  of  two  5th 
powers. 

41.  Make  up  a  binomial  the  exponents  in  whose  terms 
are  even  numbers,  and  which  can  be  factored  as  the  sum  of 


I 


FACTORING  149 


two  cubes.    Also  one  that  can  be  factored  as  the  sum  of  two 
5th  powers. 

42.  State  which  of  the  following  expressions  can  be  fac- 
tored : 

^  ^y^  a:^  +  2/^  ^  -y^^  x^-\-  y^° 

3^  +  y^  a:^  _  yi2  ^e  _p  yS  ^e  _^  yU 

3^  +  y^  x^-y^  x^-\-  y^  x^  -  y^^ 

Factor  and  check  each  of  the  following  miscellaneous 
examples : 


43. 

a:2  -.  4a2 

51. 

x^-^a^ 

44. 

o^'-8a' 

52. 

x'  -a^ 

45. 

x^  —  4:ax  +  4a2 

53. 

^12      _^      y9 

46. 

x'^  -  ^ax  +  3a2 

54. 

^12  _  2^12 

47. 

x'-a' 

55. 

6a2  -  13a  +  6 

48. 

x'  +  a' 

56. 

16x'  -  Sxy  +  1/2 

49. 

x"  -  4(a  +  6)2 

57. 

x^  +  27a^x 

50. 

a:^  -  8(a  +  hf 

58. 

x^'^  +  f 

59.  Make  up  and  work  an  example  in  factoring  to  illus- 
trate each  case  treated  thus  far. 

60.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

Case  VII 

99.  A  Polynomial  whose  Terms  may  be  Grouped  so  as  to 
be  Divisible  by  a  Binomial  Divisor. 

Ex.  1.     ax  —  ay  —  bx  +by  =  (ax  —  ay)  —  (bx  —  by) 

=  a{x  -  y)  -  b{x  -  y) 
=  {a  —  b)  (x  —  y)  Factors 

The  last  step  in  the  preceding  process  is  sometimes  clearer  to  the 
pupil  when  written  in  the  following  form: 

(x-v)  )a(i-y)-b(x-y)  ^  (^  _  ^)  („  _  j)  p^^, 
a  —  0 


15a  SCHOOL  ALGEBRA 

Ex.  2.     1  +  15a*  -  5a  -  3a3  =  1  -  3a'  -  5a  +  15a* 

=  (1  -  3a3)  -  5a(l  -  3a») 
=  (1  -  3a3)  (1  -  5a)  Factors 
Ex.  3.    a»  +  3a2  -  4  =  a'  +  2a^  +  a^  -  4 

=  a2(a  +  2)  +  (a  +  2)  (a  -  2) 

=  (a  +  2)  (a2  +  a  -  2) 

=  (a  +  2)  (a  +  2)  (a  -  1)  Factor* 

Let  the  pupil  check  the  above  examples. 

EXERCISE  42 

Factor  and  check: 

1.  ax-}-  ay  +  bx  -\- hy  14.  a:^  -  a;^  -  4x  +  4 

2.  x'^  -  ax-\-  ex  -  ac  is.  a^o^-h^x^-a^y^-^-h'^f 

3.  5a:!/  -  lOy  -  3a;  +  6  16.  a;(a:  +  4)^  +  4(a;  +  4) 

4.  3am-4mn-6a2/+8ny  17.  a2(a  +  3)  -  3(a  +  3) 
..  5.   a^x  +  3aa:  +  oca;  +  3ca:  is.  2(ar^  -  y^)  _  (3.  _  y) 

6.  3a2  +  3a6  —  5an  —  56n  19.  4a:(a;  —  Vf  -^  x  —  \ 

1.  x^^^^2^-\-2x  20.  a:^  -  1  +  2(a:2  _  1) 

8.  2a:4-2ar'-2aV+2a2a:  21.  ar^  -  y^  _|.  ^  _  ^3 

9.  2/^  +  2/^  +  y  +  l  22.  a^-y  +  ar^-y* 

10.  a^  —  2a^x  —  x-\-2a  23.  a:^  -  y^  _  ^^  _|_  ^2 

11.  x^  -{-Zy  -Zx-  xy  2^.  3?  -  f  -  {x  -  yY 

12.  ^  -  z^  -  z-^l  25.   4a2  -  a^a:^  4-  a:2  _  4 

13.  ab  -by  -  a-{-y  26.  a^^-y^+ar^-y^+a^-y 

27.  4aa:3  _j_  g^^  -  8a  -  4aa:2 

28.  a(3a  -  a:)2  -  Qax^  +  2a:3 

29.  a:^  _  8  _  7(^  _  2) 

30.  4(a:3  _!_  27)  _  313.  _  93 

31.  (2a:  +  1)3  -  (2a:  +  1)  (3a:  +  4) 

32.  (2a:  -  3)3  +  2a:2  -  9a:  +  9 


FACTORING  151 

If  33.  a:^  -  7a:  -  6 

i  34.  x3  -  Sx^  -  lOo:  +  24 

[  35.  :t^  -Sx^  +  17a:  -  10 

Factor  and  check  each  of  the  following  miscellaneous 
examples : 


36. 

a'-Sa^ 

44. 

a:^  -  a:2  +  a:  -  1 

37. 

a"  -  16a:2 

45. 

a:^  -  9a:2  +  jg^ 

38. 

a2  -  Qax  +  9x^ 

46. 

0:6  +  (a  +  ^)3 

39. 

ax  —  hx  -\-  ay  —  by 

47. 

a^  -  y^ 

40. 

a^-a-2 

48. 

(a  +  by  -  2(a  +  b)p  +  p2 

41. 

3^-a^-\-  2(x  -  a) 

49. 

(a  +  6)2  _  (a  +  6)  -  2 

42. 

3a2  -  4a  -  4 

50. 

x^-\-a^-a^-a^ 

43. 

x'  +  y' 

51. 

x  +  a^  —  a  —  a^ 

52.  Make  up  and  work  an  example  to  illustrate  each  case 
in  factoring  treated  thus  far. 

53.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

Special  Cases  Under  Case  III 

100.   By  the  Grouping  of  Terms  we  may  often  reduce  an 
'Qxpression  to  the  difference  of  two  perfect  squares. 

Ex.  1.    Factor  a:^  -  ixy  +  4i/2  -  9z\ 

a;2  _  4xy  +  4?/2  -  9^2  =  (x^  -  ^xy  +  41/2)  _  9^2 
=  (x  -  2yY  -  9^2 
=  [{x  -  2y)  +  32]  [{x  -  2y)  -  Sz] 
=  {x  -2y  +  Sz)  {X  -2y  -  Sz)  Factors 

Ex.  2.    Factor  a^  -  a:^  -  i/^  +  6^  +  2ah  +  2xy. 

o2  -  a:2  -  1/2  +  62  +  2db  +  2xy  =  {a^  +  2ah  +  62)  -(a;2  -  2xy  + 1/^) 

=  (a  +  6)2  -{x-  2/)2 
=  (a  +  fe  +  ic  -  ?/)  (a  +  6  -  a:  + 1/) 
Let  the  pupil  check  the  above  examples.  r  actors 


152  SCHOOL  ALGEBRA 

EXERCISE  43 

Factor  and  check: 

1.  a^-]-2ab  +  b^-x^  ll.  2ab -\- x^  -  a^  -  b^ 

2.  a2  -  2a6  +  62  _  4^2         ^g.   x'^ -\- a^  -  y^  -  2ax 

3.  a^  —  x^  —  2xy  —  2/^  13.   a^  -\-  y'^  —  x^  -\-  2ay 

4.  9a2  -  a:^  -  4a:y  -  4?/2      i4.   a^  -  a:^  -  2a:2!/  -  1/2 

5.  16a2  -  ar^  +  2a;!/  -  2/2      15.  a:^  _  ^/Z  _  i  _  22/ 

6.  m^  —  a;2  —  2/2  —  2x1/         16.   1  +  2xy  —  ^  —  y^ 

7.  a2  +  62  +  2a6  -  4a;2        17.  c^  -  a2  -  6^  +  2a6 

8.  a2  +  62  _  4a;2  4- 2a6        is.  a2  +  62  -  c2  -  2a6 

9.  a2  -  4a:2  +  62  +  2a6        19.  2a6  +  a262  +  1  -  a:2 
10.  a:2-2a6-a2-62  20.  222-42-22^  +  2 

21.  2O2/2  +  a:2  -  4?/2  -  2522 

22.  o2  +  2a6  +  62-c2-2c(i-(^ 

23.  a;2  +  42/2  _  9252  _  1  _  /^^y  _  52 

24.  9a2  -  25a;2  +  462  -  1  -  10a;  -  12a6 

25.  a2  -  962^2  -  1  +  66a:  -  10a6  +  2562 

Factor  and   check  each  of  the  following  miscellaneous 
examples : 

26.  ax  —  bx  -^  ay  —  by  3^.  a^  —  ¥  +  x^  —  W 

27.  a2  -  a:2  -  2x2/ -  !/2  35.  a^  +  W  -  y'^ -\- 2ab 

28.  a2  +  ax  —  a6  —  6x  36.  a^  —  27?/^ 

29.  a2  -  2a6  +  62  -  x^  37.  a^  -  Qay  +  9y^ 

30.  2a  +  26  -  3a  -  36  38.  a^x  -  16x2/^ 

31.  4a2  +  4a  +  1  -  62  39.  3a2  -  4a  +  1 

32.  9x2  -  4a2  -  4a6  -  62  40  ^^_g(^a  +  bf 

33.  9x2  _^y2^^__y  41  a^  +  2/^ 


I 


FACTORING  153 


r42.  Make  up  and  work  an  example  in  each  case  in  factor- 
ing treated  thus  far. 

43.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

101.  The  Addition  and  Subtraction  of  a  Square  will  some- 
times transform  a  given  expression  into  a  difference  of  two 
perfect  squares. 

Ex.  1.    Factor  a'  +  a%^  +  b\ 

Add  and  subtract  a%'^. 
a*  +  a262  4_  54  =  (j4  4.  2^262  +  6*  _  ^^252 
=  (a2  +  62)2  _  ^^252 
=  (a2  +  62  +  ab)  {0?  +  62  -  ah)  Factors 

Ex.  2.    Factor  ic^  -  IxY  +  2/^- 
Add  and  subtract  ^x^y"^. 
x^  -  7a;2?/2  +  2/4  =  3,4  _^  2xhf  +  y^  -  9xY 

=  (x^  +  y^y  -  9xY 

=  {x"^  +y^  +  Sxy)  (a;2  +  y^  -  Sxy)  Factors 

Let  the  pupil  check  the  above  examples. 

EXERCISE  44 
Factor  and  check: 

1.  c'  +  c'^x'  +  x'  6.  49c^  -  llcW  +  25(i^ 

2.  3:^  +  0:2+1  7.  16a:4-9a;2  +  l 

3.  4x^  -  13x2  +  1  8.  lOOx^  -  61a;2  +  9 

4.  4a4  -  21a262  +  954  9  225a^64  -  4a262  +  4 
5    93.4  _|_  33,y  _}_  4^/4  iQ  32a^  +  26^  -  56a262 

11.  a^  +  46*  12.   l  +  64a:^  13.  a:y  +  324 


154  SCHOOL  ALGEBRA 

Factor  and  check  each  of  the  following  miscellaneous 
examples : 

14.  a^  +  2a^x^ -\- a^  20.  4a'^  -  ^a^x^  +  a^ 

15.  a'  +  a^x'^  +  x'  21.  a^  -  ax  -  Qx" 

16.  a^  +  4aV  +  3a:*  22.  ¥ -^  a^b^ -{- a"^ 

17.  a^-4x^  23.  a^  +  2a^b^ -\- b^ 

18.  a^  +  4x*  24.  a^-x^ 

19.  4a^  -  Ua^x^  +  a:^  25.  a«  +  64a:« 

26.  Make  up  and  work  an  example  in  each  case  in  factoring 
treated  thus  far. 

102.  Other  Methods  cf  Pactoring  algebraic  expressions  will 
I  e  treated  later.    Thus  it  will  be  found  that 

a2  +  62  =  (a  +  6  -f  V2ab)  (a^  +  b^  -  V2ab) 
Also,       a2  +  62  =  (a  +  V^b)  (a  -  \/^6) 
Factoring  by  use   of  the  Factor   Theorem  is  treated  in 
more  advanced  text-books. 

103.  General  Principles  in  Factoring.  It  is  important  in 
factoring  to  reduce  each  expression  to  its  prime  factors. 
Therefore  it  is  important  to  use  the  different  methods  of 
factoring  in  such  a  way  as  to  obtain  prime  factors  most 
readily. 

Hence,  in  factoring  any  given  expression,  it  is  useful  to 

1.  Observe,  first  of  all,  whether  all  the  terms  of  the  expression 
have  a  common  factor  {Case  I);  if  so,  remove  it. 

2.  Determine  lohich  other  case  in  factoring  can  be  used  next 
to  the  best  advantage. 

3.  If  the  expression  comes  under  no  case  directly,  try  to  dis- 
cover its  factors  by  rearranging  its  terms;  or  by  adding  and  sub- 
tracting the  same  quantity;  or  by  separating  one  term  into  two 
terms. 


155 


I  FACTORING 

.  Continue  the  process  of  factoring  until  each  factor  can  be 
resolved  no  further, 

EXERCISE  46 


Review 


Factor: 

1.  3a;3  -  Sx  26. 

2.  2a;3  -  8x^  +  Sxy^  27. 

3.  x^  -  lla:2  +  SOx  28. 

4.  4x*  +  5xh/  —  Qxy^  29. 

5.  12a2  -  2ab  -  SOb^  30. 

6.  a;^  -  1  -  "/  +  2y  31. 

7.  40a3  -  5  32. 

8.  IQx*  -  ^Ox^y  +  25xY        33. 

9.  x^  -^Sax  -  Sa  -  X  34. 

10.  Sx"^  -  Sx  35. 

11.  4a^  -  5a2  +  1  36. 

12.  2x8  _  32  37, 

13.  a:2  +  4x  -  45  38. 

14.  4x2  +  2a  -  a2  -  1  39. 

15.  5ax^  -  5a  40. 

16.  18x3  _  3^.2  _  363-  41 

17.  x*  +  3xV  +  42^  42. 

18.  a^x^  -  9x2  -  a2  -I-  9  43 

19.  110  -  X  -  x2  44. 

20.  3x2  ^  i^jcy  _  302^2  45 

21.  7a  -  7a%*  46. 

22.  6x2  4.  14a;  -[-  8  47. 

23.  x^  -  (x  -  2)2  48. 

24.  3a  +  Sa*  49. 

25.  a^  -  a2  +  2a  -  2  50. 


6x5  _  2x  -  4x2 
1  -  23^2  +  z* 
128  -  2y^ 
1  -  a2  -  62  _  2a6 
21a2  -  17a  -  30 

3.12     _j_    yl2 

8x3  +  729^9 

405xV  -  45x4 

a5  _  4^3  ^  5^2  _  20 

(c-hdy  -1 

(x  -  yy  +  2(x  -  2/) 

24x2  +  5xy  -  36i/2 

x3  -  2x^2/  -  4x2/2  +  Sy* 

(a2  -  6)2  -  a2 

24+2:2  +  1 

(a2  -  62  -  c2)2  -  462c2 

21x2  -  40x1/  -  21t/2 

32  +n5 

5x^  +  5xy^ 

m}  +  n^ 

2ax3  +  iai/' 

1  +  X  —  x^  —  x^ 

x2  -  9  -  7(x  -  3)« 

4a4  -  37a2  +  9 

x«  -64. 


156  SCHOOL  ALGEBRA 

51.  a;3  -  27  ^  7{x  -  3)  56.  4(^2  -  52)  -  3(a  +  6)* 

52.  32a;^i/  -  2/2^0  57.  (a  -  6)3  +  (a:  -  yY 

53.  (a:2  +  j/^)^  -  16xV  58.  (a^fe  -  aV^Y 

54.  x^  +  a;2?/2  -  y'^z'^  -  ^  ^        59.  x^^  +  a» 

55.  aa:^  -  ax  —  x^y  +  y  60.  a;^  +  2/^  +  (a;  +  t/)' 

61.  (fl  -  by  {x  +y)  -\-{a  -  b)  {x  +  yY 

62.  (a  -  by  +  4(a  -  6)  (x  +  ?/)  +  4(a;  +  2/)2 

63.  ai2  -  1  65.   4a2  -962-1-66 

64.  4a2  -  962  +  4^^  _  55  gg    (3,2  _  ^^2)2 

67.  (a:2  -  1)2  +  (2a;  +  3)  (x  -  1)2 

68.  a2  -  6^  -  a^x^  +  b^^ 

69.  3a;2  _  27  +  aa;2  -  9a 

70.  a?  +  3a26  +  3a62  +  6^ 

71.  a?  -  3a2j  +  3aa;2  -  2? 

72.  a26ca;  —  amnpx  +  mhipy  —  abcmy 

73.  4a;  +  ian  +  x^  -  Aa^  -  n^  +  4t 

74.  2{x^  -  8)  +  7a;2  -  17a;  +  6 

75.  a*  -  46^  +  a2  +  262 

76.  (Sx^y  -  Sxy^y 

77.  18a;2  +  52xy  -  Qy^ 

78.  (x  +  1)'  -  x^ 

79.  (1  -  2a:)2  -  x* 

80.  aa;2  —  ex  +  ax  —  c 

85.  a;4  -  492/2  +  9 

86.  xY  -  4x2  +  4  -  2/2  -  4^22^2  _^  4py 

87.  o27ix  —  bcni^yz  +  ocmxz  —  abmny 

88.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

89.  Work  again  the  odd-numbered  examples  on  p.  88. 


81. 

7^  -  79x2  _|.  1 

82. 

a2  -  9  +  962  -  606 

83. 

x«  -  4x*  -  16x2  +  64 

84. 

(x2  +  3)3  -  64x« 

6x2 

FACTORING  157 

104.   Factorial  Method  of  Solving  an  Equation. 
Ex.  1.    Solve  a:2  +  5a:  -  24  =  0. 
Factoring  the  left-hand  member,  we  obtain 
(x  +8){x  -3)  =0 

If  any  factor  of  a  product  equals  zero,  the  entire  product  equals 
zero.  Hence  to  obtain  the  roots  for  the  above  equation,  we  may 
let  each  factor  in  the  left-hand  member  equal  zero  and  obtain  the 
value  of  X  from  the  two  resulting  simple  equations. 

Hence  we  have  for  the  above  equation 


x  -}-8  =  0 

a;  =  —  8  Root 
Check  for  a;  =  —  8 
x^  -^5x  -  24 
=  64-40-24 
=  0 


Also    a;  -  3  =  0 

X  =  S  Root 
Check  for  x  =  3 

x^  +  5x  -  24 

=  9-1-15-24 
=  24-24 
=  0 


Ex.  2.    Solve  x(x  -  2)  (3a;  -h  4)  (a;  +  1)  =  0. 

Using  the  above  method,  we  obtain 

a;  =  0,  2,  -  t,  -  1  Roots 

Check  for  x  =  0.  x{x  —  2)  (3a;  -1-  4)  (a;  -j-  1)  Let  the  pupil 
=  0(0  -  2)  (0  -t-  4)  (0  -h  1)  apply  the  checks 
=  0(-2)4xl=0  for    the    other 

values  of  x. 

Ex.  3.    Solve  7?  -x^  =  ^x-4:. 

Transposing  all  terms  to  the  left-hand  member,  we  have 

a;3  -  a;2  -  4a;  +  4  =  0 

Hence,  x\x  -  1)  -  4(a;  -  1)  =  0 

{x  ■-  1)  (a;2  -  4)  =  0 

(x  -l)(x  +2)(x  -2)  =0 

a;  =  1,  2,  -  2  Roots 
Let  the  pupil  check  the  work. 


158  SCHOOL  ALGEBRA 

EXERCISE  46 

Solve  and  check  each  of  the  following: 

1.  x^  -  5x  +  6  =  0  14.  a:2  +  2a:  =  0 

2.  0:2  _  a.  _  2  =  0  15.  a:2  +  ao:  =  0 

3.  x^  -7x  =  -  12  16.  a^  -  a^x  =  0 

4.  a:^  ~  a:  =  6  17.  a:^  +  a:^  =  4a:  +  4 

5.  a:2  =  a:  +  12  18.  a:^  +  a:^  -  9x  -  9  =  0 

6.  a:2  -  16  =  0  19.  a:^  -  5a:2  +  4  =  0 

7.  a:2  =  9  20.  a:^  -  a:2  -  4a:  +  4  =  0 

a  xix^  -  4)  =  0  21.  3(a:2  _  i)  _  2(3;  +  1)  =  0 

9.  a:^  _  25a:  =  0  22.   1  +  a:^  =  2ar^ 

10.  a:^  =  9a;  23.  y*  —  9y^  =  0 

11.  2a:2  _  33.  ^  1  =  0  24.  p2  _  3p  -f  2  =  0 

12.  3ar^  -  4a:  =  4  25.  3m^  -4m  +  l  =  0 

13.  a:^  —  a:^  —  6a:  =  0  26.  z^  —  4z  +  4  =  0 

27.  y*  -  13!/2  +  36  =  0 

Form  the  equation  whose  roots  are 

28.  3  and  4  30.-3,-7  32.  0,  2 

29.  -  5,  2  31.   1,  2,  -  2  33.  2,  3,  0 

34.  The  square  of  a  certain  number  diminished  by  4  times 
the  number  equals  45.    Find  the  number. 

35.  The  square  of  a  certain  number  increased  by  6  times 
the  number  equals  40.    Find  the  number. 

36.  What  number  plus  its  square  equals  12? 

37.  The  square  of  a  certain  number  diminished  by  9  times 
the  number  equals  zero.    Find  the  number. 


FACTORING  159 

38.  The  square  of  what  number  equals  25  times  the 
number? 

39.  The  cube  of  what  number  equals  25  times  the  number? 

40.  Find  two  consecutive  numbers  whose  product  is  72. 

41.  If  to  3  times  the  square  of  a  certain  number  we  add  4 
times  the  number,  the  result  equals  4.    Find  the  number. 

42.  The  depth  of  a  certain  lot  equals  three  times  the  front, 
and  the  area  of  the  lot  is  7500  sq.  ft.  Find  the  dimensions 
of  the  lot. 

43.  The  temperature  at  which  iron  fuses  is  2800°  F., 
which  is  332°  more  than  4  times  the  temperature  at  which 
lead  fuses.    Find  the  temperature  at  which  lead  fuses. 

44.  The  area  of  Texas  is  265,780  sq.  mi.  This  is  29,240 
sq.  mi.  less  than  6  times  the  area  of  New  York.  Find  the 
area  of  New  York. 

45.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

46.  How  many  examples  in  Exercise  30  (p.  120)  can  you 
work  at  sight? 


CHAPTER   IX 

HIGHEST   COMMON   FACTOR  AND   LOWEST 
COMMON   MULTIPLE 

105.  Utility  in  the  Highest  Common  Factor  and  Lowest 
Common  Multiple.  The  advantages  in  the  knowledge  and 
use  of  the  largest  factor  common  to  two  or  more  expressions 
and  of  their  lowest  common  multiple  are  similar  to  those 
found  in  arithmetic  for  the  same  principles.  They  aid  in 
reducing  fractions  to  a  simple  form,  in  adding  and  subtract- 
ing fractions,  and  in  multiplying  and  dividing  fractions. 
Other  advantages  will  appear  later. 

Why  do  we  now  proceed  to  make  definitions  and  rules? 

Highest  Common  Factor 

106.  A  common  factor  of  two  or  more  algebraic  express 
sions  is  an  expression  which  divides  each  of  the  given  expreS' 
sions  without  a  remainder. 

The  highest  common  factor  of  two  or  more  algebrai< 
expressions  is  the  product  of  all  their  prime  common  factors 

Thus,  the  highest  common  factor,  or  H.  C.  F.,  of  4x2,  12x',  anc 
IQx^y  is  4x2. 

107.  The  Method  of  Finding  the  H.  C.  F.  is  to 

Factor  the  given  expressions,  if  necessary; 

Take  the  H.  C.  F.  of  the  numerical  coefficients; 

Annex  the  literal  factors  common  to  all  of  the  expressions^ 
giving  to  each  factor  the  lowest  exponent  which  it  has  in  any, 
expression. 

160 


HIGHEST  COMMON  FACTOR  161 

Ex.  1.    Find  the  H.  C.  F.  of  ^x'y  -  I2xy'^  +  ^f  and  3a:y 
f  ^xjf  -  12y\ 

6xh/  -  12xy^  +  61/  =  6y(x  -  yY 

SxY  +  9xy^  -  12y'  =  Sy^{x^  +  Sxy  -  ^y^)  =  ^yKx  +  4i/)  {x  -  y) 
:.  H.  C.  F.  =  Sy(x  -  y) 

Ex.  2.    Find  the  H.  C.  F.  of  ?^a{a^h  -  ah^f  and  a\h  -  a)\ 

Zaia^h  -  ab^Y  =  Sa[ab{a  -  b)Y  =  Sa%^{a  -  by 
a\b  -  aY    =  a\  -  {a  -  b)Y  =  a\a  -  bY 
:,  H.  C.  F.  =  a^{a  -  bY 

P  EXERCISE  47 

Find  the  H.  C.  F.  of 

1.  4:a%Qab^  6.  x^  -  3x,  x'^  -  9  \ 

2.  5a^y,  15a:V  7.  Ax^  +  6x,  Ga:^  +  9a; 

3.  24aV,  56a3a:2  8.  a^  -  3^,  a^  -  x^ 

4.  24:xy,  48fla;2,  36a;  9.  xy  -y^ot^  -  x 

5.  34a5a:3^  Slrta;^  10.  ^a^  +  2a^  Aa^  -  a 

11.  a;2  +  a;,  a:2  -  1,  a:2  -  a:  -  2 

12.  ^a^x  -  ^ax?,  Sa^^  -  Sax\  ia^x\a  -  xf 

13.  2a:3  _  2x,  2x^  -  3a:,  4a;(a:  -  l)^ 

14.  ^x^  H-  ^xy  —  42/2,  ^x?  +  4iXy  —  3y^ 

15.  Ss^  -  5a:2  _  2x,  4a?  -  5a;2  -  6a;,  a:^  _  4^  * 

16.  b  -  a%  36  -  a'b  -  2a'b,  ¥  -  a'b^ 

17.  1  -  a\l  -  a^  3a  +  3a2  +  3a^  1  +  a^  +  a^ 

18.  Find  the  H.  C.  F.  of  the  numerator  and  denominator 
of  the  fraction  iii  Ex.  5  (p.  170). 

19.  Beginning  with  Ex.  17  (p.  170),  treat  each   example 
through  Ex.  22  in  the  same  way. 


162  SCHOOL  ALGEBRA 

Find  the  H.  C.  F.  of 

20.  {a'^b  -  aW)\  -  aW(a  -  by 

21.  9(x^  -  xyf,  12a;2(x2  -  y'^f 

22.  (a  +  b){x-  y),  (a  -  b)  (y  -  x) 

23.  (a  +  b)(x-  yy,  {a  -  b)  (y  -  x)^ 

24.  4.-x\x^-x-2,(2-  xy 

25.  3a2  -  10a  +  3,  9a  -  a\  (3  -  ay 

26.  x^{x  —  a)^  a*(a2  —  x^) 

27.  In  Exs.  1  and  6,  name  some  common  factor  of  the  two 
given  expressions  which  is  not  their  H.  C.  F. 

28.  Write  two  expressions  whose  H.  C.  F.  is  aV. 

29.  Write  also  two  expressions  whose  H.  C.  F.  is  Sx{x  —  1). 

30.  Write  three  expressions  whose  H.  C.  F.  is  a{x  —  b). 

31.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

Lowest  Common  Multiple 

108.  A  common  multiple  of  two  or  more  algebraic  expres- 
sions is  an  expression  which  will  contain  each  of  them  with- 
out a  remainder. 

The  lowest  common  multiple  of  two  or  more  algebraic 
expressions  is  the  expression  of  lowest  degree  which  will 
contain  them  all  without  a  remainder. 

Thus,  the  lowest  common  multiple,  or  L.  C.  M.,  of  Sa^,  Wx,  and 
4ax2  is  12a^x'^. 

109.  The  Method  of  Finding  the  L CM.  is  to 
Factor  the  given  expressions ,  if  necessary; 
Take  the  L.  C.  M.  of  the  numerical  coefficients; 

Annex  each  literal  factor  that  occurs  in  any  of  the  given  eX" 


LOWEST  COMMON   MULTIPLE  163 

pressions,  giving  the  factor  the  highest  exponent  which  it  has  in 

any  one  expression, 

Ex.  1.     Find  the  L.  C.  M.  of  Sx^  -  9^^,  a^  -  9x,  and 

x^-6x  +  9. 

Sx*  -  9x'  =  3x\x  -  3) 

x^  -9x  =  x(x  +S)  (x  -  3) 

x^  -Qx  -\-9  =  (x  -  3)2 

.  •  .  L.  C.  M.  =  SxKx  +S)(x  -  3)2 

Ex.  2.    Find  the  L.  C.  M.  of  {a'b  -  ab^f,  2ab{b  -  af,  and 

a\a^  -  62)2. 

{a'h  -  a¥y  =  [ab{a  -  b)Y  =  a?b\a  -  by 
2ab(b  -  ay  =  2ab[  -  {a  -  6)]^  =  2a6(a  -  6)^ 

^2(^2   _  ^2J2    ^    Q2(a    +  6)2  (a    _   6)2 

.  • .  L.  C.  M.  =  2a%'{a  +  by  (a  -  by  Ans. 

EXERCISE  48 

Find  the  L.  C.  M.  of 

1.  3a26,  2a62  6.   12a26,  16a62,  24a262 

2.  12a2a:2,  9ay  7.  2x{x  +  1),  a:2  -  1 

3.  2aCySbc,4ab  8.  3a2  +  3a6,  2a6  +  26^ 

4.  3a26,  4ac2,  662c  9.   7x^  2x^  -  Qx 

5.  42a:y,  2Syh^  10.  a:^  -  1,  a;2  -  1 

11.  x^  —  1/2,  0^2  —  Sxy  +  2!/2 

12.  3x3  _  3^^  5^2  _  12a:  +  6 

13.  5aa:2(ar  —  yY,  3bxy(x^  —  y^) 

14.  :t^  -Sx^-  40a:,  a:2  -  9x  +  8 

15.  a2  —  62,  a^  —  6^  a^  +  6^ 

16.  6a:2  +  6a:,  2a:3  _  2x2^  3a;2  _  3 

17.  4a26  +  4a62,  6a26  -  6a62,  3a2  -  362 

18.  2a:2  +  a;  -  1,  4a:2  -  1,  2a:2  +  3a:  +  1 

19.  30:^  -  3,  6a:2  _  i2a;  +  6,  20:^  ^  2a:2  +  2x 


164  SCHOOL  ALGEBRA 

20.  12a:3  _  2x^  -  140a;,  \M  +  6a;  -  180,  ^x^  -  39x2  ^  53^ 

21.  1  —  a:  4-  a:^  —  a:^^  ;[  _l_  ^  _l_  ^2  _|_  ^^  2a;  —  2x^ 

22.  {x  -  1)3,  7a:!/3(ar^  -  1)2,  \^x^y{x  +  1)^ 

23.  ISa:^  -  12a;2  +  2a:,  27a:^  -  ?>x',  \%^  -  24^2  _^  g^. 

24.  (a:  -  1)  (a:  +  3)2,  (x  +  1)2  {x  -  3),  (a:2  -  1)2,  ar^  _  9 

25.  Find  the  L.  C.  M.  of  the  denominators  of  the  fractions 
in  Ex.  18  (p.  181). 

26.  Find  the  L.  C.  M.  also  of  the  denominators  of  the 
fractions  in  each  example  from  Ex.  21  to  Ex.  28  (inclusive), 
p.  181. 

Find  the  L.  C.  M.  of 

27.  (a^h  -  a¥)\  aW{a  +  6)2 

28.  {abc  -  body,  (3a^c  -  3acd)^  Qa^c^  -  6aW 

29.  (a26  -  a62)2,  (^2  _  ^5)2^  (^3  ^  ^2^)2 

30.  9(ar^  -xy)\  12(a:2  _  ^^2^2^  jg^^  ^  ^2^)2 

31.  a  —  6,  6  —  a 

32.  9(a  -  6)2,  12(6  -  a)2 

33.  (a  +  6)  (a:  -  2/),  (a  -  h)  {y  -  x) 

34.  (a  +  6)  (a:  -  y)\  {a  -  h)  {y  -  xf 

35.  4  -  a:2,  a:2  -  a:  -  2,  (2  -  a:)2 

36.  a:2(a;  —  of,  x(a^  —  x^), 

37.  Find  two  consecutive  numbers  the  difference  of  whose 
squares  is  5. 

38.  Make  up  and  work  an  example  similar  to  Ex.  37. 

39.  The  reclaimable  swamp  land  in  the  United  States  and 
the  land  that  is  capable  of  irrigation  equal  178,000,000  acres 
all  together.  If  the  irrigable  land  exceeds  the  swamp  land 
by  22,000,000  acres,  how  many  acres  of  each  of  these  kinds 
of  land  are  there? 


LOWEST  COMMON   MULTIPLE  165 

40.  The  distance  from  New  York  to  Havana  is  1410  mi. 
If  a  steamer  leaving  New  York  travels  at  the  average  rate 
of  260  mi.  per  day,  and  one  leaving  Havana  at  the  same  time 
travels  at  the  average  rate  of  280  mi.  per  day,  how  many 
days  and  hours  will  elapse  before  the  two  steamers  meet? 

41.  The  distance  of  the  sun  from  the  earth  is  92,800,000 
mi.  This  distance  exceeds  107  times  the  diameter  of  the  sun 
by  95,200  mi.    Find  the  diameter  of  the  sun. 

42.  A  man  bequeathed  $20,000  to  his  wife,  daughter,  and 
son.  To  his  daughter  he  left  $2000  more  than  to  his  son,  and 
to  his  wife  three  times  as  much  as  to  his  son.  How  much 
did  he  leave  to  each? 

43.  The  distance  of  the  moon  from  the  earth  is  238,850 
mi.  This  exceeds  110  times  the  moon's  diameter  by  1030  mi. 
Find  the  diameter  of  the  moon. 

44.  If  10  m.  exceeds  10  yd.  by  33.7  in.,  how  many  inches 
are  there  in  a  meter? 

45.  Write  a  common  multiple  of  the  expressions  in  Ex.  1, 
which  is  not  their  L.  C.  M. 

46.  Write  a  common  multiple  of  the  expressions  in  Ex.  10 
which  is  not  their  L.  C.  M. 

47.  Write  two  expressions  whose  L.  C.  M.  is  24a^6V. 

48.  Write  two  expressions  whose  L.  C.  M.  is  12a:^(a;  —  2)^ 
(X  -  1). 

49.  Make  up  and  work  an  example  similar  to  Ex.  27.  To 
Ex.31.    To  Ex.  47. 

50.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

51.  How  many  examples  in  Exercise  35  (p.  131)  can  you 
work  at  sight? 


CHAPTER   X 

FRACTIONS 

110.  TTtility  of  Fractions.  In  algebra,  as  in  arithmetic, 
fractions  are  useful  in  indicating  new  units,  and  in  indicating 
quotients  and  thus  opening  the  way  to  save  labor  by  cancel- 
lation. 

In  algebra  fractions  also  have  other  uses  besides  those 
which  appear  in  arithmetic.  Thus,  in  algebra,  a  fraction  is 
often  useful  in  expressing  a  general  formula. 

Ex.  If  an  automobile  goes  a  miles  in  h  hours,  how  far 
would  it  go  in  c  hours  at  the  same  rate? 

Y  =  no.  of  miles  the  automobile  travels  in  1  hour 

0 

—  =  no.  of  miles  the  automobile  travels  in  c  hours 
b 

Why  do  we  now  proceed  to  make  definitions  and  rules? 

111.  A  Fraction  is  the  indicated  quotient  of  two  alge- 
braic expressions.    This  quotient  is  usually  indicated  in  the 

form  -. 
b 

The  fraction  -  is  read  "a  divided  by  6,"  or,  for  brevity, 
b 

"a  over  b." 

Note  that  the  dividing  line  of  a  fraction  takes  the  place  of 

a  parenthesis  and  is  in  effect  a  vinculum. 

166 


FRACTIONS  167 

Another  method  of  writing  the  preceding  fraction  is  a/6.  This  is 
called  the  solidus  notation.  It  is  convenient  in  printing  mathematical 
expressions,  and  is  much  used  in  European  mathematical  hterature. 

X  4-  1 

— — —  written  in  the  solidus  notation  would  be  {x  +  l)/{Sx  —  5) 

3x  —  5 

The  numerator  of  a  fraction  is  the  dividend  and  the  de- 
nominator is  the  divisor  of  the  indicated  quotient. 

Terms  of  a  fraction  is  a  general  name  for  both  numerator 
and  denominator. 

EXERCISE  49 

1.  If  three  boys  weigh  a,  b,  c  pounds  respectively,  what  is 
their  average  weight? 

2.  If  four  boys  can  run  the  quarter  mile  in  p,  q,  r,  s  sec- 
onds respectively,  what  is  their  average  time? 

3.  How  many  acres  are  there  in  a  field  a  feet  long  and  h 
feet  wide? 

4.  How  many  acres  are  there  in  a  field  c  rd.  X  ^  rd.?  In 
one/ yd.  X  e  ft?    p  ft.  X  g  rd.? 

5.  If  sugar  is  worth  a  cents  a  pound,  how  many  pounds 
can  be  obtained  in  exchange  for  b  pounds  of  butter  worth 
c  cents  a  pound? 

6.  If  coal  is  worth  c  dollars  a  ton,  how  many  tons  of  coal 
can  be  obtained  in  exchange  for  p  tons  of  hay  worth  b  dollars 
a  ton? 

7.  Make  up  and  work  a  similar  example  concerning  c 
calves,  worth  a  dollars  each,  exchanged  for  chairs  worth  d 
dollars  each. 

8.  If  coal  is  worth  c  dollars  a  ton,  how  many  tons  can  be 
obtained  in  exchange  for  /  bushels  of  wheat  worth  h  cents  a 
bushel  and  for  w  bushels  of  corn  worth  y  cents  a  bushel? 


168  SCHOOL  ALGEBRA 

9.  Who  first  used  the  letters  a,  6,  c  to  represent  known 
numbers?    (See  p.  268.)    Tell  all  you  can  about  this  man. 

10.  Before  the  use  of  a,  b,  c,  what  other  symbols  were  used 
to  represent  known  numbers?  Discuss  the  relative  advan- 
tages in  these  different  sets  of  symbols. 

11.  As  a  notation,  in  what  respects  is  a/b  superior  to 

a  H-  6?    To  --?    In  what  respects  is  it  inferior  to  each  of  these? 
b 

12.  How  many  examples  in  Exercise  45  (p.  155)  can  you 
now  work  at  sight? 

112.  An  integral  expression  is  one  which  does  not  contain 
a  fraction;  as  3x^  —  2y. 

An  expression  like  5x^  +  fa;  +  }  in  which  fractions  occur  only 
in  the  numerical  coefficients  is  sometimes  regarded  as  an  integral 
expression. 

A  mixed  expression  is  one  which  is  part  integral,  part 
fractional. 

Thus,  3^2  +  X  -  5  +  A^ 
'  3x2-2 

113.  Sign  of  a  Fraction.  A  fraction  has  its  own  sign,  which 
is  distinct  from  the  sign  of  both  numerator  and  denomina- 
tor. It  is  written  to  the  left  of  the  dividing  line  of  the  fraction. 

The  sign  of  —  t  is  — ,  and  the  sign  of  -^^  is  +  understood. 

0  0 

General  Principles 

114.  A.  //  the  numerator  and  the  denominator  of  a  fraction 

are  both  multiplied  or  divided  by  the  same  quantity,  the  value  of 

the  fraction  is  not  changed. 

For  if  a  dividend  is  denoted  by  D,  its  divisor  by  d,  and  the  quo- 
tient by  Q  T) 

~  =  Q,  Siiid  D  =  d  X  Q 
a 


I 


TRANSFORMATIONS  OF  FRACTIONS  169 


If  m  denotes  any  multiplier,  Dxm=dXmX  Q,  or 

^=Q  (Art.  15,  p.  18) 

Also  if  m  denotes  any  divisor  except  zero, 

D^m  =d  ^mXQ,  or  f  "^  ^  =  Q  (Art.  15) 

a   -j-  m 

115.   B.  law  of  Signs.    By  the  laws  of  signs  for  multipli- 
cation and  division  (see  Arts.  50,  62,  pp.  59,  77), 
a— aa— aaa  a  a 


b      -})        h         b         -V  be  -bXc       -bX  -c 

x  +  y ^      x  +  y      ^  _  X  +  y 
y  -  X      -  {x  -  y)  X  -  y 

Or,  in  general, 

The  signs  of  any  even  number  of  factors  of  the  numerator 
and  denominator  of  a  fraction  may  be  changed  without  changing 
the  sign  of  the  fraction. 

But  if  the  signs  of  an  odd  number  of  factors  are  changed,  the 
sign  of  the  fraction  mu^st  be  changed. 

Transformations  of  Fractions 
I.  To  Reduce  a  Fraction  to  its  Lowest  Terms 

116.   A  Fraction  in  its  Lowest  Terms  is  a  fraction  whose 
numerator  and  denominator  have  no  common  factor. 
To  reduce  a  fraction  to  its  lowest  terms,  as  in  arithmetic, 
Resolve  the  numerator  and  the  denominator  into  their  prime 
factors,  and  cancel  the  factors  common  to  both. 

Ex.  1.    Reduce  ,^  ,  ,  „  to  its  lowest  terms. 

Divide  both  numerator  and  denominator  by  12a^x^  (see  Art.  114). 
.      SQa^x^         Sa 


^Sa^x^y^      ^y^ 


Ans, 


170  SCHOOL  ALGEBRA 

E     2     Q^^  -  12^^       36(3a  -  46)  ^  36    . 
"^^    •    12a2  -  16a6       4a(3a  -  46)  ~  4a 

Notice  particularly  that  in  reducing  a  fraction  to  its  lowest  terms 
it  is  allowable  to  cancel  a  factor  which  is  common  to  both  denomina- 
tor and  numerator,  but  it  is  not  allowable  to  cancel  a  term  which 
is  common  unless  this  term  is  a  factor. 

Thus,  —  reduces  to  -  ; 

ac  c 

but  in  ,  a  of  the  numerator  will  not  cancel  a  of  the  denominator. 

a  +y 

This  is  a  principle  very  frequently  violated  by  beginners. 

EXERCISE  60 

Reduce  each  of  the  following  to  its  simplest  form: 
^    27  g     3a'  -  6a'b         ^^    45(a:  -  yY 


36  4aW  -  8a¥  18(a;  -  yY 

2    108  g         2a  ^g      a^b  +  ab^ 


144  -  4a2  -  2a  2a^b  -  2ab'' 


150  Gax  -  \2ay  ^x'y  -  \2xy^ 

My^                ^^      4x-h4y  ^^    6aW-\-12a¥ 

11.    18. 


12aV  iax  +  4ay  9a'b  +  18a262 

g    12a:V  a:^-y^  .^     2a;^  -  Sxy 

15s^y^^  '   (x  +  2/)2  •  4a:3  _  g^y2 

^         Sa^x  13    12aV  -  Sa^xy  ^       49x^  -  64^^ 

^   127?f^  ^^    8{7?  -  1)  2^         ^-21 


mxy^'^  \2x  -  12                   a:2  -  6x  +  9 

22  (3;  -  y)^  (a:  +  y)^  24  6a;^  -  a^y  -  2^^^ 

'          (x2  -  2/2)3  •  6a:2  _  7^^  _j_  22^2 

23  20:^  -  8y2  ^5.  ^^  +  ^)'  "  ^' 


4a:2  _  2xy  -  12y^  a^  -  (6  +  c)^ 


TRANSFORMATIONS  OF  FRACTIONS  171 

26.     l-C"-'^)'  29       ^-^^ 


a^-(i 

X-l)2 

12a;2- 

2oa;  - 

24a2 

4a:2_ 

2ax  - 

6a2 

a^- 

-8 

30. 


31. 


x^  -  :(?  -  6a:2 
a:^  —  2/^ 


x'  +  xy  +  2/^ 

ax  —  bx  —  ay  -j-  by 


x^  —  z^  —  4:  —  2xy  —  4:Z  -\-  y^ 

32. 


x^y^  +  2a:!/2  +  4?/^ 

x^  —  4  —  22/2  —  4a:  +  !/^ 

1+4 
What  is  the  correct  value  of  the  fraction  ^    .    ^  ?    If 
6  +  4 

the  4's  are  struck  out,  what  does  the  value  of  the  above 

fraction  become?    Is  it  allowable,  therefore,  to  strike  out  the 

4's  in  the  above  fraction? 

34.  Make  up  and  work  an  example  similar  to  Ex.  33. 

35.  Which  of  the  following  fractions  can  be  simplified  by 
striking  out  the  4's? 

1+4         a;  +  4  4a;  3X4  4a  4a 

11  +  4        2/  +  4        4(1  +  2/)      4  +  11        a;  +  4        4a; 

36.  Make  up  and  work  an  example  similar  to  Ex.  35, 
involving  3's. 

37.  Which  of  the  following  can  be  simplified  by  striking 
out  the  62's? 

b^-\-x        b^         b^-4  a^b^  o^ 

b^  +  y        b'^y        36^  +  4        a^  +  6^        6V 

38.  Which  of  the  following  can  be  simplified  by  striking 
out  a  +  6  in  both  numerator  and  denominator? 

a  +  6  5(a  +  b)  3x(a  +  b)  3(a  +  b) 

a  +  b  +  c         3(a  +  6)  +  c         4y{a  +  b)         5(a  +  6)  +  c 

39.  Make  up  and  work  an  example  similar  to  Ex.  38  con- 
cerning the  striking  out  of  x.    Of  (p  +  q)^. 


172  SCHOOL  ALGEBRA 

40.  Why  is  it  allowable  to  subtract  4  from  each  member 
of  the  equation  a;  -h  4  =  a  +  4  and  not  from  each  term  of 

the  fraction ? 

a  +  4 

41.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

EXERCISE  61 

(x  —  2Y 

1.  Reduce  ^^- ^  to  its  lowest  terms. 

4  —  or 

(x  -  2y  ^  {x  -2)(x-  2)  ^  (2  -x){2-  x)  ^  2  -x 
4-x^        {2-i-x){2  -x)       {2+x){2  -x)       2+x 

Check.    Let  a:  =  1,  then,  ^f  ~  ^]'  =  1-=^  =  1 
'  '4-^2         4-1        3 

.,  2-^2-11 

^^'  2T^=2-iri  =  3 

Reduce  to  simplest  form  and  check  the  work: 
(a  —  by  ^      a  -{-  b  —  c 


3. 


4. 


5. 


6. 


b' 

-a" 

(2a- 

-yY 

f 

-^ 

9-  to'' 

m^ 

-  7to  +  12 

9- 

-r' 

(.X- 

-3)^ 

2- 

■  y 

-4 

a  — 

h 

h  —  a 


c2  -  (a  +  bf 

9 

6?/ -3a: 

12ai/  -  6aa; 

10. 

6a26  -  3a3 

4a262  -  8a63 

n 

a:3-27 

9  -  6x  +  ar^ 

19 

4  -  (a  +  6)2 

(a  -  2)2  -  62 

T» 

ax  —  bx  —  ay  -]-  by 

TRANSFORMATIONS  OF  FRACTIONS  173 

Without  changing  the  value  of  the  fraction 

14.  Change  each  of  the  following  so  that  the  denominator 
of  the  fraction  shall  be  a  —  b. 

-3  3  X         X-  y         -  Sx      2a  -  36 

b  —  a         b  —  a        b  —  a      b  —  a        b  —  a        b  —  a 

15.  Change  each  of  the  following  so  that  the  denominator 
shall  be  {x  —  y)  {x  —  y). 

3  -4  a-b 


(y  -  x){y  -  ^)           (y  -  x)  (x  -  y)  {y  -  x) 

16.  Show  that — r  equals 


{y  -x){z-  y)  {x  -  y){y  -  z) 

17.  By  changing  signs  of  factors,  write  each  of  the  follow- 
ing in  three  different  ways: 

5  a  —  b  _  a  —  b  a  —  b  (b  —  a)  {c  —  d) 

a-b      X  -  y      c  -  d      (x  -  y)  (y  -  z)      {y  -  x)  {y  -  z) 

Solve  the  following  equations,  after  reducing  the  fraction 
in  each  equation  to  its  simplest  form: 

18.  ^^^  +  2a:  =  5  20.  ^^^^  =  7  +  ar^ 

x  —  I  a:  +  3 

19. -  a:^  =  7  21. —  =  b  —  X 

a:  —  1  x^ 

^^    ax  —  Za  -\- bx  —  Zb      ^  ^ 

22.      ; =    17 


23. — ^±JIli^  =  15  -  3a; 


a  +  6 

ax  - 

-  5a  -  6a:  +  56 

a-b 

a^  - 

-8       a^  +  S 

24. -  ~ — -^  =  20 

a: -2         a: +  2 

25.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 


174  SCHOOL  ALGEBRA 

II.    To  Reduce  an  Improper  Fraction  to  an  Integral  or 
Mixed  Quantity. 

117.  An  Improper  Fraction  is  one  in  which  the  degree  of 
the  numerator  equals  or  exceeds  the  degree  of  the  denomi- 
nator. 

Since  a  fraction  is  an  indicated  division,  to  reduce  an  im- 
proper fraction  to  an  integral  or  mixed  expression, 

Divide  the  numerator  by  the  denominator; 

If  there  is  a  remainder,  write  it  over  the  denominator,  and 
annex  the  residt  to  the  quotient  with  the  proper  sign. 

3^  +  4x^-5 


Ex.    Reduce 


x^-j-x  +  2 


x^  +4x2  -  5 

x^  +    x2  +  2x 


x^  +x  +2 
X  +S 


Sx^  -2x  -  5 
Sx^  +  3a;  +  6 
-  5x  -  11 

=  X  +S Ans, 


x^  +x  +2  a:2+x+2 

When  the  remainder  is  made  the  numerator  of  a  fraction  with 
the  minus  sign  before  it,  as  in  this  example,  the  signs  of  terms  of 
the  remainder  must  be  changed,  since  the  vinculum  is  in  effect  a 
parenthesis  (see  Art.  41,  p.  50). 

EXERCISE  52 
Reduce  each  of  the  following  to  a  mixed  quantity: 
«    121  ^    181 

2.    3.    

9  17 

^    10a^3^  +  5ax  -7  -  a 


1. 

32 
5 

4, 

0:2- 

-2x  +  S 

X 

ti 

4a:3  +  6a:  -  5 

7. 


5ax 


2x  x-h  1 


I 


TRANSFORMATIONS  OF  FRACTIONS  175 

g    x'  +  ?.xy-  2y^  -  1         ^^        9a' 


3a:*  -  13x  - 

28 

+  2-  a 

X  — 
ar*+l 

1 

x^  +  x-1 

a;2  -  a:  -  1 

2x^  +  7 

^^-{-x+l 
x^-\-x^  -  X 

-  1 

16. 


10.    -^ 17. 


11.    -i: ■ 18. 


3a2  -  26 

a:^  +  a;^  -  4a;  +  7 
a:  +  3 

2a2 


a-{-b 

0^  -  x!^  -h  3^  -  2x 
a^  +  1 


12.  — — ^^ 19. (To  three  terms.) 


13.   — ■ 20. 


14. t: 21 


1  -\-x 

1 


l-\-x- 
8 


a:2  +  2  2  +  a:-a:2 

22.  Make  up  an  improper  fraction  with  a  monomial  de- 
nominator and  reduce  it  to  a  mixed  number. 

23.  Make  up  an  improper  fraction  with  a  binomial  de- 
nominator and  reduce  it  to  a  mixed  number. 

24.  How  many  examples  in  Exercise  1  (p.  8)  can  you 
now  work  at  sight? 

III.  To  Reduce  a  Mixed  Expression  to  a  Fraction 

118.  To  Reduce  a  Mixed  Expression  to  a  fraction,  it  is  nec- 
essary simply  to  reverse  the  process  of  Art.  117.    Hence, 

Multiply  the  integral  expression  by  the  denominator  of  the 
fraction,  and  add  the  numerator  to  the  result,  changing  the  signs 
of  the  terms  of  the  numerator  if  the  fraction  is  preceded  by  the 
minv^  sign; 

Write  the  denominator  under  the  result. 


176  SCHOOL  ALGEBRA 

g^  +  y' 
Ex.    x  +  y-  ^^ 

^  (a;  +  y)  (a^  -  y)  -  {^'^  +  j/^) 
a:  -  1/ 

a;2    _  2/2    _  a;2    _   ^2  _   2^2 


a;  -  2/  X  -  2/ 


Ans. 


I 

Reduce  to  a  fraction 


y  -  X 

EXERCISE  63 


1.  3^  .    2.   12|  3.   13iV 

4.    a  -  1  +  -  10.    a  -  a:  +  1  -  ^-^ 

a  a  -\-  X 

-±-  11.  ?^^ 

X  —  1  a  —  2 


5.    a:  +  l+^L_  11.    ^?LZ^^a-l 


6.    a:2  +  a;-l ^  12.    a:  -  a  -  ^^^^ ^' +  y 

a;  —  1  X  -\-  a 

,.   4.-2-f^  13.    1-^--^  +  "' 
2a:  +  1  26c 

7.-262  ,.    A    J.  (2  -  3a)2 
a  +  26  4 

16.    M_li)!  +  2a6 
4 


IB  TRANSFORMATIONS  OF  FRACTIONS  177 

IB  20.  The  distance  from  New  York  to  Chicago  is  912  mi., 
which  is  100  mi.  more  than  one  fourth  of  the  distance  from 
New  York  to  San  Francisco.    Find  the  latter  distance. 

21.  A  running  horse  with  a  rider  has  gone  1  mi.  in  1  min. 
35^  sec,  which  is  13 J  sec.  more  than  three  times  the  time  in 
which  an  automobile  has  gone  one  mile.  Find  the  latter 
time. 

22.  Make  up  two  mixed  numbers  of  your  own  and  reduce 
them  to  improper  fractions. 

23.  How  many  of  the  examples  in  this  Exercise  can  you 
work  at  sight? 

IV.    To    Reduce    Fractions   to    Equivalent    Fractions   of 
THE  Lowest  Common  Denominator 

119.  To  Reduce  Fractions  to  their  lowest  common  denomi- 
nator, as  in  arithmetic,  we 

Find  the  lowest  common  multiple  of  the  denominators  of  the 
given  fractions; 

Divide  this  common  multiple  hy  the  denominator  of  each 
fraction; 

Multiply  each  quotient  hy  the  corresponding  numerator;  the 
results  will  form  the  new  numerators; 

Write   the    lowest  common   denominator   under   each  new 

numerator, 

2       3  5 

Ex.    Reduce  - — ,  ^-t-,  and  - — -  to  equivalent  fractions 
6ax  4:a^x  oax^ 

having  the  lowest  common  denominator. 

The  L.  C.  D.  is  12aV. 

Dividing  this  by  each  of  the  denominators,  we  get  the  quotients 
^ax,  3x,  and  2a. 


178  SCHOOL  ALGEBRA 

Multiplying  each  of  these  quotients  by  the  corresponding  numer- 
ator and  setting  the  results  over  the  common  denominator,  we  obtain 

Sax        9x        10a      , 

Ans. 


!  12a^x^'  12aH^'  12a%2 

i 

'  EXERCISE  64 

Reduce  the  following  to  equivalent  fractions  having  the 
lowest  common  denominator: 


1. 

5    7 

8' 12 

3    4    9 

2. 

5^  15'  20 

2x  5x 

3. 

9'  6 

12a    7    a 

4. 

56 '  10'  b 

1      2     1 

b. 

2a¥'  a^V  ah 

6. 

2      3           1 
3a2'  4aa:'  ^""^  x 

ac  ab  be   ad 

'    M'^'^'Yc 

s.     /     ,2,     % 
a^  —  a       a  —  1 

3         ^        1    1         1 

1  +  x'    '  a;'  x  +  or^ 

10         ^            1 

^°  ^_1'^_1 

11     ^      11 

"'    4x2  _  9'  2a;  +  3'  X 

1 

'x 

1+x         1-x 
2-2x'^'3  +  3;r 

13. 

3 

4           4 
l'x2  +  a.+  i'^ 

14. 

3               4 

a%  +  a¥'  a^h  -  ab^ 

15. 

1 

5           3 

3a:- 

-  6'  2x  +  4'  x2  -  4 

ifi 

2 

X                   X 

a;  -  a:^'  3  _|_  3^'  2  -2ar 


i      ADDITION  AND  SUBTRACTION  OF  FRACTIONS   179 
Processes  with  Fractions 
I.  Addition  and  Subtraction  of  Fractions 

120.   The  Method  of  Adding  or  Subtracting  Fractions^  as  in 
arithmetic,  is  to 

■Reduce  the  fractions  to  their  lowest  common  denominator; 
Add  their  numerators^  changing  the  signs  of  the  numerator  of 
any  fraction  preceded  by  the  minus  sign; 
Set  the  sum  over  the  common  denominator; 
Reduce  the  result  to  its  lowest  terms. 

Ex.  1. -  a  +  -. h  - 

a  —  1  a^  —  a      a 

a  a  1  1 

"1       Zo  ~       ' 


a  —  1       1       a^  —  a      a 

a^  -  a^  +0^  -\-l  +a  -1 

a{a  —  1) 
-  a'  +  2a2  +  a       -  a^  +  2a  jM  ^^ 


a{a  —  1)  a 

Ex.  2.    Simplify  -^  +,     "^ 


x''  -1       x-hl       I  -  X 

The  factors  of  x^  —  1  are  x  -{-  1  and  x  —  1.  Hence,  if  the  sign  of 
the  denominator,  1  —  a:,  is  changed,  it  will  become  x  —  1,  and  be  a 
factor  of  a:2  -  1.  But  by  Art.  115  (p.  169),  if  the  sign  of  1  -  x  is 
changed,  the  sign  of  the  fraction  in  which  it  occurs  must  also  be 
changed.    Hence,  we  have 

x"^  X  X      _  x"^  +  x^  —  X  +  x^  -\-  x  _     Sx^       . 

x^  -l'^  x  +  1'^  ^"^  ^~^  ^^^ 

Where  the  differences  of  three  letters  occur  as  factors  in  the  va- 
rious denominators,  it  is  useful  to  have  some  standard  order  for 
the  letters  in  the  factors.  It  is  customary  to  reduce  the  factors  so 
that  the  alphabetical  order  of  the  letters  is  preserved  in  each  factor,  . 
except  that  the  last  letter  is  followed  by  the  first.  This  is  called 
the  cyclic  order.  ^  I  j 

Thus,  a  —  6,  6  —  c,  c  —  a  are  written  in  the  cyclic  order. 


180  SCHOOL  ALGEBRA 

Ex.    3.    Simplify 


(a  —  b)  (c  —  a)       (a  —  b)  (c  —  b)       (c  —  b)  (a  —  c) 

Changing  c  —  b  io  b  —  c,  and  a  —  c  to  c  —  a  where  they  occur, 
we  obtain 

1  1.1 


(a  -  6)  (c  —  a)       {a  -  b)  {b  -  c)       {b  -  c)  {c  -  a) 
-  b— c— c-{-a+a— b 
{a  —  b)  {b  —  c)  (c  —  a) 
^  2a -2c  ^  -2  , 

\  (a  -b)(b  -  c)  (c  -a)       (a  -  6)  (6  -  c) 


EXERCISE  66 


Find  the  value  of 


3. 


,3,21  ^       a             h 

1. 1 —    D. —   r- 

2x      X      '^x  a  —  b      a  -\-  0 

„2         3,1  ,           3a; +  1,1- 3a; 

3a      ^ax      X  8               6 

J 2__l  Q    g  +  l      g-l 

2ac      3a6      6c  '22 

a  +  26  _  6a-  1  ^    a;  -  1  _  a;  +  1 

■      2a6            6a2  "  a;  + 1      x  -  1 

^    2a2a;  +  3  ,   ,      3a  +  a;  ,^    x  +  1      ^   .  7-3a; 

3a  —  46  2a  —  6  —  c  ,  15a  —  4c 

2x^y  —  33  _  arz^  —  i/^g      y  —  3x3^  _  2 

^^'       Sx^y  2xy^             Qxh         3 


a:-2      a;  +  2  (m  -  1)^      m^  -  1 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS     181 
(a +  6)2         26  ,^      3a:  2a;      ,     lOx 


4(a-6)2      a-6  a:  +  2      a:-2'ar^-4 


a:^  +  a:  a;^  —  a; 

1  1        .    a: 


3x-3      2a; +  2      6a:2_6 


2a:  -  1      4a;  +  2      4x^-1 

a:       ,^      a:  —  1      a:  —  2 
20. h  2 

a;2  -  1  ^         a:  +  1      a;  -  1 

3  4,2 

21.    —— —  + 


22. 


23. 


a:  +  l      a;  +  2      a;  +  3 

a:  +  2  a;-3.        2a: +  5 


2a;2  +  a:  -  1      4a;2  -  1   '  2a;2  +  3x  +  1 

6      _      a6       _      ab^ 
^T6      (a  +  6)2  ~  (a  +  6)^ 


24.      2a:2/      J   3y   ^   3a:      3a;2  -  Sy^ 
a;2  —  i/2      2a:      2?/  2a;i/ 

25    ^^~y  _L     l^^y     _  3a; +  2/ 
a;  +  22/      a;2  —  4i/2      x  —  2y 

2       ,  3a:  -  1      2a:  -  5 

26.   1 +  X  — 


27. 


28. 


a;  —  1  a;  +  1 

2  a:-3 


a;  +  4      a;2  -  4a:  +  16      a;^  +  64 
5x  7  26 


'  2(a:-3)2      3a:  +  9      4a;2  -  36 

Reduce  each  of  the  following  fractions  to  its  lowest  terms 
and  collect: 

3? -9     x-i  x^-l^         (x  +  iy 


182  SCHOOL  ALGEBRA 

31       46^         (a-hy      g      3a +  36 
•   a2  -  62       a2  -  62  "^         3a  -  36 

3a:      ,       4       ,       1 
32.   -.^ 7  +  ;^ + 


0:2  -  1   '   1  -  a;      1  +  a: 
33.       2a  -1      ^     2 


a^  —  b^      a  +  b      b  —  a 
34.       3a:t/  y  -  a:  ^  y  +  a: 


Q?  —  4y^      2y  -\-x      2y  —  X 


a:  -  1      1  +  a:  '   1  -  r^ 
36.  ^^!±^'-_^+      2/ 


a:2-2/2      a:  +  i/      t/-a: 

^n        '3        ,       5  7a 

37. h 


8 -8a      4a +  4      Sa^  -  8 
38.3+^-+     5-  1  3 


39. 


X      X  —  1      1  —  x^      x-\-l      a:  +  a:^ 
1  1 


(a:  -  2)  (3  -  a:)       10  -  7a:  +  a:^      (5  _  3.)  (^  __  3) 


.n  2  3^4 


+ 


(a  -  3)  (6  -  2)       (a  -  2)  (2  -  6)       (a  -  2)  (3  -  a) 
5 


(a  -  3)  (2  -  6) 

26  +  a      26  -  a      46a:  -  2a2 


41 


x-^  a        a  —  X  x^ 


42     ^  +  1         2a:-l  2 7^ 

^    '   6a:-6      12a:  +  12      3  -  3^^      12a: 

a:^  -  a:  -  6       a:^  +  4a:  +  3  _     15a: 
a:2  4-  5a:  +  6      a:^  -  4a:  +  3      9  -  x" 
a^  +  2a6  +  6^  _        4a2  -  6^  a^  -  2a6  +  36^ 

a2  -  62  2a2  -  3a6  -  262  "^  a^  -  3a6  +  262 


43. 


MULTIPLICATION  OF  FRACTIONS  183 

45. z  + 


9-0:2  ■  3(^_j_3)2        5(x-3y        5x2-45 
^^        3a:  +  2       ,  x  4  -  x 


a:2  -  5a:  +  6      8a:  -  a:2  -  15      7a:  -  a:^  -  10 


(a  —  6)  (a  —  c)      (6  —  c)  (6  —  a)       (c  —  a)  (c  —  6) 


{a  —  b)  (a  —  c)       (b  —  c)  (b  —  a)       (c  —  a)  {c  —  b) 

49.       y  +  ^       I       ^  +  ^       I       ^  +  y 


(a:  -y)  (x-  z)      (y  -  z)  {y  -  x)      {z  -  x)  {z  -  y) 

50. y^ + ?^- -  +      ""y 


{x-  y)  {x  -z)       {y  -  z)  {y  -  x)       {z  -  x)  (z  -  y) 

SJ"     71 TTl T  +  Z T^ t:   + 


52. 


(I  —  m)  {I  —  n)       (m  —  n)  {m  —  I)       (n  —  /)  (n  —  m) 

>    y     i     ^  r     l-a:  1     1       1?  ^ 

a:       ^a:+l       La:2_3,_|_i      a:4-lJ  )       3:^  +  1 

53.  Make  up  and  add  three  fractions  with  monomial 
denominators. 

54.  Also  three  with  binomial  denominators. 

55.  How  many  examples  in  Exercise  2  (p.  13)  can  you 
now  work  at  sight? 

II.  Multiplication  of  Fractions 

121.  The  Method  of  Finding  the  Product  of  two  or  more 
fractions,  as  in  arithmetic,  is  to 

Multiply  the  numerators  together  for  a  new  numerator,  and 
multiply  the  denominators  together  for  a  new  denominator,  can- 
celing factors  that  are  common  to  the  two  products. 

This  method  reduces  the  multiplication  of  fractions  to 
the  multiplication  of  integral  expressions,  and  enables  us  to 
use  again  our  knowledge  of  the  latter  process. 


184  SCHOOL  ALGEBRA 

Ex.    ^+J^xf~^lx      ^ 


-*:+  2/  ^  (x  + 1/)  (x  -  y)  ^^ 


X  a:(a;2  +  y^)       ^  (aL±4/)  (a;  ^y) 

4:(x  -  y) 


a;2  +?/2 


Ans. 


II.  Division  of  Fractions 


122.  The  Method  of  Dividing  one  fraction  by  another  is 
the  same  as  in  arithmetic.    For 
a      c      aXd      b  X  c 


b '  d      bXd ' bXd 
aXd 


(see  Art.  114,  p.  168) 


©(^^ 


bXc 

Hence,  to  divide  one  fraction  by  another, 
Invert  the  divisor  and  proceed  as  in  multiplication, 

x{x-{-l)      x^-\-x+l  '   {x  +  1)2 
_  (x  -l){x^  +X  +  1)       {x^  -  1)  {x^  -  1)       (x  +  ly- 

X{X  +  1)  ^  X^  +X  +  1  ^   (X   -   1)3 

Ans. 


{x  +  ly 


X 

The  reciprocal  of  a  number  is  the  result  obtained  by  divid- 
ing unity  by  the  given  number. 

Thus,  the  redprocal  of  2  is  1  -J-  2  or  ^;  oi  x  la  -. 

z  x 

Hence,  the  reciprocal  of  a  a  fraction  is  the  fraction  inverted. 

2  2.33 

Thus,  the  reciprocal  of  ^  is  1  -^  q)  that  is,  1  X  qj  or  ^. 

„.    .,    ,      ,         .  .    .a .    h      ^a  —  h  .   X  +  y 

Similarly,  the  reciprocal  of  -r  is  -;  of  ^         is  , . 


L 

H  1    ^x^y 


DIVISION  OF  FRACTIONS  185 

EXERCISE  56 


bx^y       2SaW  ^  a^b^  +  Sab  ^  ab-\-3 

Ua'c       15xy^  '     4a^  -  I     *  2a  + 1 

21xf^28^  7  a;^-9  .   a;  -  3 

^     U^    '  39s4  '  x^  +  x  '  x^-l 

9a%  ^  28ax'^  .  21a^x  (a  -  If  ^  x  +  1 


ScPx      15b^c       lOftc'  o(a;  + 1)^      (a  -  1)^ 


492/"  40a:3  9a:2  -  1      i2a:  -  18 

15a;        ^  2a:(x  +1)     ,^    2x' -  x  -  I  ^^x^  -  \ 
2a:(2a:  -  1)  ^x"  2a;2  +  a:  -  1       a:^  -  1 

a^i/  —  ax^y    .  a^i/  —  2aa:^  +  x'^y 


a^x^  -\-  a^x^y  a?  +  ay 


12.    ^ — t-tt:  -^  ^ — t-tt;  14.       a;  H X 


(a +1)2      (a+1)'  \        a;-l/      x=  +  l 

3x^+x-2^      4^-J  /a;        \   .   /x^ 


2/^' 


16.  ^X^^X^^X^^ 
5a:i/         x-\-y      {x  -  yf      2xy 

^^    3(a  -  6)2      7(a2  -  62)   ^      i4a& 


4(a  +  6)2'^  9(a-6)3      8(a  +  6) 
a;2  +  2a:  -  3      a;2  +  2a;  -  15  .  a:^  +  5ar^ 
a;2  4-  a;  -  12       a;2  +  2a;  -  3    *  a;^  +  4a;2 
^^    6a;2y  -  4a;i/2  ^  30a;  +  20y  ^     xy 

45a;2  —  202/2  4a;22/2  x-\-y 

2^    6a;2  -  5a;  -  4       6a;2  +  a;  -  2       2a;2  +  5a;  -  12 
*  2a;2  +  7a;  -  4      4a;2  -  4x  -  3       9^2  -  6a;  -  8 

a;«-l 


3X.(.  +  1  +  1)(.-1  +  1). 


a;2(a;2  -  1) 


186  SCHOOL  ALGEBRA 

«a-)(-Dx(-^) 

23.         ^  +  2/"        X  fl  +      ^    ^   .  'a:'  -xy-\-y^ 
'  3^  +  x^y  -i-xy^      \        X  -  yj   '       x^  -  f 

'   a'-{x-\-iy      l-{a-xy  '  a-x-1 

25    2-6-aa^-62-46-4       5^-a^-46+4 
*   6-2-aa2  +  62  +  2a6-462_^2^4^_4 

\a6      6c      acj\b      c      a)  '        a^H^c^ 

3s,   (\  +  l,  +  l-l)4^±^(lz^x(l+?  +  a)] 
\a^      a:^ax/[_  ox  \        x        J  _] 

Vm  -\-  2n      m  —  27i~|  _^  Vm  -{-2n  _m  —  2?i~| 
L^Ti  —  2n      m  +  2/1 J    '    L^  —  2/1      m  +  2n J 

30.  Write  the  reciprocal  of  each  of  the  following:  3,  a,  2x, 
4  1    a_  J^  a  +  2a;        1 

5'  5'  2a;'  2x  a  -  2V  a"  -  2b' 

31.  Make  up  and  work  two  examples  involving  both  mul 
tiplication  and  division  of  fractions. 

32.  How  many  of  the  examples  in  Exercise  15  (p.  60)  can 
you  now  work  at  sight? 

IV.  Reduction  of  Complex  Fractions 

123.  A  Complex  Fraction  is  one  having  a  fraction  in  its 
numerator,  or  in  its  denominator,  or  in  both. 

In  simplifying  any  complex  fraction,  it  is  important  to 
write  the  entire  fraction  at  each  step  of  the  process. 


29 


I 


REDUCTION  OF  FRACTIONS  187 


X.1     -^ ^=;,X-^   =   ^^^«*. 


; 


y       y 

When  the  numerator  and  denominator  of  a  complex  frac- 
tion each  contain  fractions,  the  expression  is  often  simplified 
most  readily  if  we 

Multiply  both  numerator  and  denominator  by  the  lowest 
common  denominator  of  the  fractions  contained  in  them, 

i  +  i  +  i 

Ex.2.    Simplify    ^ — ^ — -■ 

-  +  y  +  i 

y      z       X 
Multiplying  both  numerator  and  denominator  by  xyz,  obtain 

m  xH  +  xy^  +  yz^ 

124.  A  Continued  Fraction  is  a  fraction  whose  denomina- 
tor is  a  mixed  expression,  having  another  mixed  expression 
in  its  denominator,  and  so  on  until  the  fraction  ends. 

1 


Ex.    Simplify 


x-\- 


1  - 


2 

1 


1  ,         1        )  ,  X -2 

1  X  -h 


x"^  -  4x  -2      x^  -4x 


Ans. 


X  —  5 
Hence,  in  general,  to  simplify  a  continued  fraction, 
Reduce  the  last  mixed  expression  in  the  fraction  to  an  im- 
proper fraction  (see  the  brackets  in  the  examples) ; 


188  SCHOOL  ALGEBRA 

Then  invert  the  last  fraction  and  multiply  it  into  the  numer- 
ator under  which  it  is  placed  (see  the  brace) ; 

Thu^  alternately  reduce  a  mixed  number  and  invert  a  divisoi 
fraction  until  the  simplification  is  completed. 


Simplify: 


1 

2x 

4^2 

1 

1  - 

'¥x 

1 

3 

2 

X 

0^3 

x' 

(-i) 


EXERCISE  67 


4  1,1 

--  X  x-  -  1 -— 

? 3.     ^  5.  ^+^ 

2  X  a  —  1 

«       1                         ab       ^^  2 

2  -  -  2d  X-  ~ 

i  4.    ^ -  6.   §. 

.       1  2cd  6 

a  — 


0:2  5 


11. 


J/ X 

a;         1  —  X 


12. 


1  +  a:  a; 


9  _  ^  X  1  —  X 


1  -\-  X  X 

9.  ^^   y^.  13.      °  +  i 

a  —  1 


a  ,  a: 

10.         ^       °'  14. 


\a      0      c  J 


J.  _  _1_      J.  c2(a  +  6)2  -  a^l 

n  I  o 


a26V 


I 


REDUCTION  OF  FRACTIONS  189 

5  +  «_2-l  21.   2a-l-  "-^ 


.^    a      X  ax  o        ^ 


a;      a      2  ,    1  a 

+  —  a 


a      X      a      ax  ,   1  +  a 

,.  2(i^  -  i)     1  ,  1 

16-    ^^irr~^  22.    3 


1  .-         2x 


I 


l_(a6-c(^)2  ^^^     3^ 


17    (a6  -  1)^  -  c'cP  x  +  1 

{cd  +  1)^  -  a^6^  _, 

(ab  +  cdy-l  23.    l-^4~ 

a:-l 


18.    3--^^  a  +  b 

ox 


3 


a  —  b 


24.    1 


a? 
1  „  8 


'^-fT-a  ^+2+4 


i+-    ^ 


26. 


g      6  +  c  ^  A    ,  6^  +  c^  -  g^ 

1  1      ^  V    "^         26^        y 

g      6  +  c 

4 
a: 

27.    ; —  X 


a:  +  2              a;  —  2  oif 

28. 


1 1  +  -^ 

'      1  +  ^-^  1- 


1 -X  1+X 


190 


29. 


30. 


xy^  +  x^y 


SCHOOL  ALGEBRA 


^^  ~  U  ^  c  y    w  h^^U      ~c)        \l^~c^h)' 

i_  _  /I    1 Y    1^  _  /I  _  1 Y    / 1  _  1 Y  _  i" 

62       V«       ^y        c2       \a      h)        \a       cj       b' 


1  -2 


31. 


1  -  2x 


1+2 


1  +  2x      |(i  +  a;  +  i;c2)  -  I 


1  -2a; 
Find  the  value 

r.*      22        ,  2 

32.   Of  :; —  when  v  =  -• 

l-\-2v  7 


33.    Of 


-22         ,  1 

-^-^,when.=  --. 


34.  Of  i^  when  F  =  — ,  m  =  10.2,  t)  =  5,  and  r  =  ^• 

35.  Make  up  and  simpHfy  a  continued  fraction. 

36.  How  many  examples  in  Exercise  19  (p.  78)  can  you 
now  work  at  sight? 

EXERCISE    68 

Oral  Review 


1.  Give  the  value  of  each  of  the  following: 


(1)  1-1. 


(2)  i  -  1. 


(3)l+&+£. 


(4) 


a  -^b      a  —b 


a      0 


a      0 


a.E^a,.,e.l)-    (.)g-2)'    (3,(1-D' 


PROCESSES  WITH  FRACTIONS  191 

3.  On  the  foot  rule  show  the  meaning  of  i  in.  -^  2.    Of  J  in.  4-  2. 

4.  Divide  each  of  the  following  fractions  by  2:    |,  f,  i,  f,  f, 
2a    a^  a  a  +b  6  7  3^  5^  462  362 

V  26'  6'      2    '  a'  a'  26'  46'   a  '  a  * 

5.  Divide^  by  2.    By  2a.    36.    46. 

6 

6.  Divide  1  by  each  of  the  fractions  -,  -,  -,  ^,  ^r- 

7.  Give  the  reciprocal  of  3,  -,  4,  tj  t,  r;  -  • 

8.  Give  the  value  of  -  when  ^  =  ^-     When  ^^  =  q'  !»  o'  o'  o'  5» 

__2    _1   6 
3'       a'  c* 

9.  State  the  value  of  -  when  x=  -.   When  x  = -,  -,  — ,  — ,  —, 

""  3'  2'  5 

15  1 

10.  Give  the  value  of  ;r when  x  =  —  -. 

2  +  X  5 

-11     wu^'    l-4„^-334a-la4al      2a      1« 

11.  What  IS -of-?    Of-,   ^,   ^,    -2-,   ^,   y,    2"'    -3,   4^? 

12.  If  4  is  subtracted  from  both  numerator  and  denominator  of 
■<fij,  is  the  value  of  the  fraction  changed?    By  how  much? 

13.  If  a  =  -  -  and  6  =  -  -,  state  the  value  of  -r-.   Of  -  tt- 
2  2  6  46 

14.  What  is  the  value  of  1  ^  2/3?    Of  1  4-  a/6?    Of  2  -^  x/2y? 

15.  Simplify  those  of  the  following  fractions  which  can  be  reduced 
to  lower  terms : 

4x  4a;       4a       4       a  —  6    a  -f  6    a^  +  6^  d^b 

4a  +6'  4a  +46'  46'  a  +  4'  6  -  a'  a^  +  ¥'  a^  +  x^'  a^x 

Give  the  value  of 

_  -3-1        -2  8         _8^   x"  -\ 

8       '   -  4  -  6'   -  2/3'  a/6'  \  -  x' 

17.   Of  ia'  4-  \(^    Ja'  ^  Ja2     ^x^  4-  Jx^     Jx'  ■¥  \x      1-5-  Jx« 
1  -5-  §x2    2a2  ^  fa    2  -^  5/6a; 


192  SCHOOL  ALGEBRA 

18.  Make  up  an  illustration  to  show  the  value  of  5  X  0  (for 
instance,  in  connection  with  a  pupil's  mark  for  6  examples  which 
he  failed  to  work  correctly). 

19.  Give  in  the  briefest  form  the  product  of  (a  —  b)  (b  —a).  Of 
(x  -  2y)  {2y  -  x).     Of  (a  -  b)  {a  +  b)  (a^  +  b^)  (a*  +  ¥)  (a*  +  6«). 

20.  Does  7—^3  =  -  ,     ^    J 

{y  -  xy  {x  -  yY 


EXERCISE  69 

Written  Review 

1.  Indicate  by  a  parenthesis  that  2a  —  36  +  c  is  to  be  subtracted 
from  5a  +  26  —  3c.    Then  remove  the  parenthesis  and  simplify. 

2.  Subtract  the  sum  oi^x  ■\-2y  —  z  and  a  —  x  —  Zy  from  —  5. 
Also  from  0. 

3.  Write  by  inspection  the  value  of  [(3a  —  6)  —  c  +  2d\^, 

4.  Factor  {a  +b  -  cY  -  {x  +  y  -  xY  ^  \-  -\  .  I  -f 

x^       y*  5 

1       '  1      '  1 

5.  Change  — h  -;;  so  that  it  shall  be  a  perfect  square. 

x^      xy      2/2 

6.  What  is  the  difference  between  an  exponent  and  a  power? 
Give  an  illustration. 

7.  Subtract  (5a  +  1)  (2a  -  3)  from  (a  +  2)  (a  +  1)  +  (a  +  2)^. 

8.  Find  the  value  of  3(a:  -  l)^  -  3(x  +  1)  (x  +  2)  -  x{x  -  2) 
{y  —  2x)  when  x  =  —  2  and  2/  =  —  5. 

9.  Find  the  value  of ^B —  when  A  =  5a  and  B  =  2a. 

10.  By  factoring  find  the  roots  of  a;"  -  5j  +  6  =  0.    Prove  your 
answer. 

oil        ,1    ,  f^  ~  b  .  .  ^    b  —  a 

11.  bhow  that -\  IS  equal  to  -^ * 

12.  If  a  =  12i,  6  =  37^,  c  =  33  J,  and  d  =  10,  find  in  the  shortest 

4i(P  4:(P  4c^ 
way  the  numerical  value  of  each  of  the  following:  — ,  — ,  -r-- 

13.  From  Y.OSa^  take  -  4Ja2. 

(^2    _^  52)  2 

14.  Reduce  -  1  +  — j-^ —  to  an  miproper  fraction. 


PROCESSES  WITH  FRACTIONS         193 

15.  When  we  change  a;-3=5tox=5+3,  what  is  the  change 
called?  What  right  have  we  to  make  this  change?  Why  do  we 
transpose  —  3  instead  of  adding  3  to  each  member  of  the  given 
equation? 

16.  What  is  the  use  or  advantage  in  being  able  to  find  the  H.  C.  F. 
of  two  given  expressions?  In  being  able  to  find  their  L.  C.  M.? 
Illustrate. 

3.7.  Show  that  the  sum  of  two  numbers  (as  of  a  and  h),  divided  by 
the  sum  of  their  reciprocals,  equals  the  product  of  the  given  numbers. 

18.  If  s  =  :; ,  find  the  value  of  s  when  a  =  2  and  r  =  —  7:. 

1  —  r'  2 

2  3 

Also  when  a  =  —  ^  and  t  =  —  -7. 

If  s  =  r,  find  the  value  of  s  when  r  =  jj,  Z  =  -77,  and 


W 


1 

a  = 


2' 

20.   If  a  =  3,  which  is  greater,  .^  _     or 


3a    • 

21.  Divide  2a^  +  10  -  16a  -  39a2  +  15a*  by  2  -  5a^  -  4a. 

22.  Give  an  illustration  to  show  why  3x0  gives  zero.  Also 
why  ^  gives  zero. 

23.  Show  that  a  common  factor  of  any  two  algebraic  expressions 
is  also  a  common  factor  of  their  sum  and  difference.  Of  the  sum 
and  difference  of  any  multiples  of  the  given  expressions. 

24.  Prove  that  if  half  of  the  sum  of  any  two  numbers  (as  of  a 
and  b)  is  added  to  half  their  difference,  the  result  will  equal  the  greater 
of  the  two  numbers.    Illustrate  by  two  numerical  examples. 

25.  Prove  that  if  half  the  difference  of  any  two  numbers  is  sub- 
tracted from  half  their  sum,  the  result  will  be  the  smaller  of  the  two 
numbers. 

26.  Write  an  example  of  a  continued  fraction  and  reduce  it. 

27.  Why  is  it  allowable  to  change  both  minus  signs  to  plus  in 
—  X  =  —  3,  and  not  in  —  a;  —  3? 

28.  Collect  in  a, short  way  ^^-^  +  j—^  +  ^-33  +  ^^^f^- 
Sua.  Collect  the  first  two  fractions  first. 


194 

29.  Collect  in  a  short  way 

30.  Also 


SCHOOL  ALGEBRA 
3  2 


X  +2      X  -S      X  -2   '  X  +3' 
2xy  4^y^ 


X  —  y      X  +  y      x^  +  y^      x*  +  y* 
Simplify: 

mx'  +  ^x  -  2) 


31 


33. 


34. 


35. 


36. 


t(ix2  +  ^x-i) 

x^  +  xy  —  2?/2 


32. 


X  +2  6x 


2  -X 


1  -  2x     2a;  +  1      4x2  -  1 


4x  -3 


1  "^  x  -  2      X  -S^  {x''  -x){x  -2)' 


'      4--^ 


1       '  1 

a  —  -      a  +  -      a 

a  a  a 


+ 


1     '     .       ^ 

-3     «^+;; 


+ 


+ 


1 


3x  +  2    '   {x  -1){S  -x)    '   (2  -x){x  -  3) 


37.    (^+^-2)(^+^+2).(^-^y. 
\x      a        /  \x      a        /       \x      a/ 

^     ll+a:^       1  +a:'J    "^  U  +x3       1  +  x*l 


39. 


40. 


41. 


-^ 

2 

X 

■+11 

i'4 

"-a 

(M)- 


rc  -4  - 


a;  — 


-2 


X 


a;  -4 


a;  -4 


a;  — 


X  -  5 


9^      ^ 
a:2  +9 


a;2      9  J 


X 


e-i)(i+>)  • 


42. 


PROCESSES  WITH  FRACTIONS        195 
1  +  8x»        1  -  27x' 


1-  -^^^   1+    '^ 


••■11  o^  ••• 


1  -  2a;  1  +  3x 

a:^  -  5x'  +  4      a;      a;  -  2  ^  x 

*^'    a;2  +  i     ^       1  ":;   r' 

a;  a;2 


44.  Given  a  +  b  +  c  =  2s, 

show  that  a  +b  —  c  =2{s  —  c)  and  that  a  —  6  +  c  =  2(s  —  6). 

45.  Also  show  that 

a^  +0^  -  62      2(s  -  a)  (s  -  c) 


I 


2ac  ac 

6.  Also  show  that  1  + ttt =  — ^: 

2ab  ab 

7.  Show  that 


(2a  -  Sxy  [Sx'ja  +  2xy  +  5x*  (a  +  2a;)^]  +-27x^  {a  +  2xy  (2a  -  3a;)» 

(2a  -  Sxy^ 
reduces  to 

2ax^  (24a;  +  5q)  (a  +  2a;)^ 

(2a  -  3a;)io 

48.  The  distance  from  New  York  to  San  Francisco  by  way  of 
Cape  Horn  is  13,800  mi.  This  is  1920  mi.  less  than  three  times  tho 
distance  from  New  York  to  San  Francisco  by  way  of  Panama. 
Find  the  latter  distance. 

49.  Make  up  and  work  an  example  similar  to  Ex.  48,  using  the 
fact  that  the  distance  from  London  to  Bombay  by  way  of  the  Cape 
of  Good  Hope  is  11,220  mi.,  but  by  way  of  the  Suez  Canal  is  6332 
mi. 


CHAPTER   XI 
FRACTIONAL  AND   LITERAL   EQUATIONS 

125.  A  fractional  equation  is  an  equation  that  contains 
an  unknown  number  in  a  denominator. 

Ex.    -  +  5  =  3x. 

X 
Equations  containing  binomial  numerators  and  numerical  de- 
nominators are  frequently  termed  fractional  equations,  since  they 
are  solved  in  the  same  manner  as  fractional  equations  proper.    See 
Ex.  1.  of  Art.  126. 

An  integral  equation  is  an  equation  which  does  not  con- 
tain an  unknown  number  in  a  denominator. 

126.  The  Method  of  Solving  a  Fractional  Equation.     If  an 

equation  contains  fractions,  it  is  necessary  first  to  multiply 

the  members  of  the  equation  by  such  a  number  as  will  remove 

the  fractions. 

^     ,      CI  1      a;  +  1      2a;  -  5       llx  +  5      a;  -  13 
Ex.1.    Solve  ^ 5-  =  -To S^ 

The  L.  C.  D.  of  the  denominators  is  30. 
Multiplying  both  members  of  the  equation  by  30  (see  Art.  70, 
3),  we  have 

15(x  +  1)  -  6(2x  -  5)  =  3(lla;  +  5)  -  10(a:  -  13) 
Hence,  15x  +  15  -  12x  +  30  =  33a;  +  15  -  10a:  +  130 

\hx  -  12a;  -  33a;  +  lOo;   =  -  15  -  30  +  15  +  130 
-  20a;  =  100 

a;  =  —  5  Root 

^            a;  + 1      2a;  -  5       -5  +  1       -10-5  ^   ,  o      i 

Check.  — 5"  =  ""2 ^-=-2+3  =  1 

llo;  +  5  •    a;  -  13       -55+5       -5-13  _       f-   ,  ^  _  . 

^-10 3~  =~"10 3  -5+6-1 

196 


FRACTIONAL  AND  LITERAL  EQUATIONS  197 

.X.  2.      Solve   -— h  ^ :; -^  =  0. 

1  -\-  X      1  —  X      1— ar 


Multiplying  by  the  L.  C.  D.,  1  -  x^, 

4(1  -x)  +{x+iy  -x^-^S  =0 
4  -  4x  +  aj2  +  2a;  +  1  -  x2  +  3  =  0 

-2x  =  -  8 
a;  =  4  i2oo< 
Let  the  pupil  check  the  work. 

Hence,  in  general, 

Reduce  each  fraction  in  the  equation  to  its  lowest  terms; 

Clear  the  equation  of  fractions  by  multiplying  each  member 

by  the  L.  C.  D.  of  all  the  fractions; 

Complete  the  solution  by  the  methods  of  Chapter  VI. 


I 


EXERCISE  60 

Solve  and  check  each  result: 

1. 

X      3a;      7a;      34 
3       5        5       15* 

2. 

2a;-3      a;  +  l      5a;  +  2 
4^6             12 

3. 

1       3       5  _  3  _  19^ 

2a;      a;      3a;      4a;      24 

4. 

2a;      2a;  + 1       1 
3            5           3 

5. 

3a;  +  5             a;  +  4 
4                     6 

6. 

7  =  f  (a;  -  2). 

7. 

2a;  -  8  -  j(24  -  2a;)  =  0. 

.     8. 

l(x  -  1)  =  iix  -  2). 

9-   3(fx-i)(ia;  +  f)=a^. 


198  SCHOOL  ALGEBRA 


3- 2a;      a;-3      ,      a;  +  4,l 
^°—8 6--'  =  ^-  +  2i- 

3a;  -  1      x  +  1      4x  +  1  _  3(a;  -  1) 
11.   -^  g  2i  4  ^• 

2a;  +  5      x+lj  _,_     _  5a;  -  10^      1 
"■       5  10     "^''~       20  5' 

13.  ^-!(5  +  :r)+^-i(2a;  +  5)=^^. 


2(a;  +  i)  +  a;(l-l) 


14.   2(a;  +  *)  +  a;(l-^)=^^  +  ^. 


x  +  5_a;  +  7x+l_2a:-5  ^  a;  +  22 
^^■7.52  10     ~     70    ■ 

16.  f  (5a;  +  2)  -  f  (7a;  -  2)  +  i(3a;  -  2)  =  x  -  ?• 

17.  .5a;  —  Ax  =  .3. 

18.  1.5a;  —  5  =  a;. 

19.  .6a;  -  1.5  =  .2  -  .15a;. 

1.5a;-1.6_3.5a;-2.4       24.   g^TTs  =  4^qr4- 

1.2  .8       * 

,.^«.       »     .    .  X      x^  —  5x      2 

3.2a;-3.4      .6a;  +  4  25.   -- 


23. 


5 

8 

2a;- 
6a;- 

1 
5 

3a; +  1 

_8a;-7 

21. 

4.5 

22. 

a;-l      3 
a;  +  l      5 

27. 

2.5 


3       3a;  -  7       3 

5      ,       6 
26.    , h 


1  —  a;      1  +  a;      1— a; 


-2 


3  4         8a; +  3 


28. 


3-a;      3  +  a;      9  -  x^ 

2a;  +  1  10      _^  2a;  -  1 

2a;  -  1      4a;2  _  1  ="  2a;  +  l' 


29.    — 7-^  + 


a;  +  l      a;-l      a;  +  2 


FRACTIONAL  AND  LITERAL  EQUATIONS  199 

30.  ^:zl  =  _?_+    1-3^ 


a^-S      x-2      x^  +  2x  +  4: 
31.   a;^  -  a:  +  1  _  o^  __  x^-\-x-\-l 


32. 


X  —  1  x-h  1 

x-S       jx^  +  l  _  a;  +  3 
2a:  +  6       x^  -  9       Sx  -  9 


33.    _3_  +  ^ L_  =  ^l_  + 


34. 


a:-la:  +  l      2a: -2      3a: +  31 
a:  +  la:^  +  7_^       2  a:-l 

2a: -3      4a:2  -  9      2a:  +  3      6  -  4x" 


35.   ??lzJ___L__2  ^ 


36. 


3a:  -  6      6a:  +  12  2a:2  -  8 

6a; +  6 2a:  +  1  2a: 

2ar^  +  5a:  +  3      2a:2  -  a:  -  1  ~  a:^  +  2a:' 


Reduce  each'  fraction  in  the  following  to  its  lowest  terms 
and  then  solve: 

^'•-2  =  ^-  39.  ^^^  =  4-6(.-3). 

Find  the  value  of  the  letter  in  each  of  the  following:  - 

41.   _i_  4-  I =  ?.   43.   J^ ? 14  =  0 

«  +  2      3(u  +  2)      3  3-25      6-45 

42    __1_  +  1=^_44        4  31     ^1 

3(p-7)      6      2p-14  2^  +  2      3^  +  3      6* 

..    r  +  6      2r  -  18  ,  2r  +  3      .i   ,  3r  +  4 
*'•   ^i 3-+"4-  =  ^^  +  '-12- 

46.   If  A  =  Iw,  A  =  600  and  w  =  20,  find  the  value  of  I 

Do  you  know  the  meaning  of  this  process  in  arithmetic  in  conr 
nection  with  the  rectangle? 


200  SCHOOL  ALGEBRA 

47.  In  like  manner,  find  /  when  ^  =  80  and  w  =  11^. 

48.  If  F  =  Iwh,  V  =  720,  /  =  10,  and  w  =  Q,  find  h. 

Do  you  know  the  meaning  of  this  process  in  arithmetic  in  con- 
nection with  the  study  of  volumes? 

49.  In  like  manner  if  F  =  .3Q,w  =  .8,  and  h  =  .9,  find  I. 

50.  lip  =  br,p  =  9  and  b  =  45,  find  r. 

Do  you  know  the  meaning  of  this  process  in  arithmetic  in  con- 
nection with  the  subject  of  percentage? 

51.  If  p  =  hr,  p  =  760  and  r  =  .05,  find  6. 

52.  If  i  =  prt,  i  =  $66,  p  =  $440  and  t  =  3,  find  r. 

Do  you  know  the  meaning  of  this  process  in  arithmetic  in  con- 
nection with  the  subject  of  interest? 

53.  If  i  =  prt,  i  =  $66,  p  =  $360,  and  t  =  3|,  find  r. 

54.  If  i  =  prt,  i  =  $15.75,  p  =  $75,  r  =  .06,  find  t 

55.  If    C  =  |(F  -  32)    find    F   when    C  =  50.      When 
C  =  100. 

Do  you  know  the  meaning  of  this  process? 

56.  If  LW  =  lwsindL  =  S,W  =  100,  and  w  =  40,  find  L 
Can  you  find  out  the  meaning  of  this  formula  and  process? 

57.  lfR  =  ^^,  find  s  when  R  =  10  smd  g  =  32. 

58.  If  F  =  (l  +  -^\,  find  V  when  F  =  20  and  ^  =  13. 

59.  If  Z  =  ih(b  +  6'),   K  =  280,   h  =  12,  and  b  =  10, 
find  b\ 

60.  If  F  =  TrUm,  V  =  1540,  TT  =  V-,  and  E  =  7,  find  H. 

61.  If    r  =  7ri^(i?  +  i),    T  =^  1144,    TT  =  \S    R  =  14, 
findL. 


I  FRACTIONAL  AND  LITERAL  EQUATIONS     .     201 

62.  Make  up  and  work  an  equation  containing  fractions 
with  the  denominators  4,  6,  and  12.  Can  you  form  the  equa- 
tion so  that  the  root  shall  be  1?    2?    4? 

63.  Make  up  and  work  an  equation  containing  fractions 
with  the  denominators  a:  +  2,  a;  —  2,  and  x^  —  4. 

64.  Work  again  Exercise  24  (p.  99). 

127.  Special  Methods.  The  work  of  solving  an  equation 
may  often  be  lessened  by  using  some  special  method  or  device 
adapted  to  the  peculiarities  of  the  given  equation. 

First  Special  Method.  If  in  a  given  equation  the  denomi- 
nators of  some  fractions  are  monomials,  and  of  others  are  poly- 
nomials, it  is  best  to  make  two  steps  of  the  process  of  clearing 
the  equation  of  fractions:  (1)  remove  the  monomial  denom- 
inators and  simpUfy  as  far  as  possible;  (2)  remove  the  re- 
maining polynomial  denominators. 

^     ,     ^  .       2a:  +  8J       13a:  -  2    ,  a;      7x      a:  +  16 
Ex.1.    Solve  -^ _____  +  _____. 

Multiplying  by  36,  the  L.  C.  D.  of  the  monomial  denominators, 
8a;  +  34  -  ^^^jf^.'gl^  +  12a;  =  21x  -  x  -  IQ 

Transposing  all  terms  except- the  fraction  to  right-hand  side, 
36(13a;  -  2) 


17a;  -  32 


-50 


Dividmgby-2,       17-^  ,32    =  ^5 

234a;  -  36  =  425a;  -  800 
191a;  =  764 
a;  =  4  Root 
Let  the  pupil  check  the  work. 

Second  Special  Method.  Before  clearing  an  equation  of 
fractions,  it  i^  often  best  to  combine  some  of  the  fractions 
into  a  single  fraction. 


202  SCHOOL  ALGEBRA 

X  —  1      X  —  2      X  —  3      X  —  4 


Ex.  2.    Solve 


X  —  3      X  —  4:      X  —  5 


In  this  equation  it  is  best  to  combine  the  fractions  in  the  left- 
hand  member  into  a  single  fraction,  and  those  in  the  right-hand 
member  also  into  a  single  fraction,  before  clearing  of  fractions.  We 
obtain 

-1 -1 

(x  -2){x-S)~  {x-  4)  {x  -  5) 
7 
Clearing  and  solving,  ^  =  o  ^^^^ 

Let  the  pupil  check  the  work. 

EXERCISE   61 

Solve  the  following  and  check  each  solution: 
1    ^^  ~  ^  _i_     4^         X  -j-  5 


2. 


6  3a: +  2  2 

3-2x  .  X       l-6x       2-Sx 


3.   2J- 


6      15  -  7a:  9 

2a:  —  1      X 


2a:  +  4  4  2 


^    5a:  +  13  _  2a:  +  5  23  5  -  jx 

12  6  4a: -36  3      * 

5        5     _  2ia:  -  3  _  a:  +  11  _■   11a:  +  5  _  ^ 

'   7-x  4  8  16 

g    3a:-  1      4a:-  7  ^  x  _    2a:  -  3        7a:-  15 

30  15  4      12a: -11  60 

7     6a:  -  7  _  a:  +  1      2a:  -  1  _  199 
'    11a:  4- 5         15  30  10* 

-    ^,     a:  +  4        4- 3a:,  4a: +  9      4-a;,5 
^7a:  +  ll  8      ^12  24        4 

3    3a:  -  2^  _  7         x-j    _  2a:  -  1|      2a:  +  3| 
9  12"^|a:+ll  3  6      ' 


I         FRACTIONAL  AND  LITERAL  EQUATIONS  203 

Q    2r2a:_lf3a:-l      2a:-5         n       19^^_ 


^^    1.2a;  -  1.5      Ax  +  1  ^  .4a;  +  1 
1.5  .2a; -.2  .5 

1111 


12. 


13. 


14. 


15. 


o;  —  2      x  —  3      X  —  4:      x  —  5 

X  —  1 _x  —  3  _x  —  5  _x  —  7 ^ 

X  —  2      X  —  4:      X  —  6      X  —  S 

X  —  7_x  —  8_x  —  4_x  —  5 

x  —  8      x  —  9      x  —  5      a;  —  6 

3  2               2                3 


3a;  -  2      2a;  -  3      2a;  +  3      3a;  +  2 


16.*  2^  +  1   .  2^  +  9  _  2a; +  3      2a;  +  7 

a;  +  l        a;  +  5        a;  +  2        a;  +  4  * 

^7    4a;  -  17      10a;  -  13      8a;  -  30      5a;  -  4  _ 
*a;-42a;-3         2a; -7        a;-l 

18.  Work  Ex.  2  by  clearing  all  denominators  at  once.  Then 
work  the  same  example  by  the  method  of  Art.  126.  About 
what  fraction  of  the  work  is  saved  by  the  second  process? 

19.  Treat  Ex.  13  in  the  same  way  as  you  treated  Ex.  2. 

20.  On  the  average,  the  distance  one  must  go  below  the 
surface  of  the  earth  to  get  an  increase  of  1°  in  temperature, 
is  62  ft.  This  is  1  ft.  more  than  one  third  the  distance  one 
must  go  above  the  earth's  surface  to  get  a  decrease  of  1°  in 
the  temperature.    Find  the  latter  distance. 

21.  Who,  so  far  as  we  know,  first  invented  transposition 
in  solving  equations,  and  when?  Who  first  brought  the  use 
of  transposition  into  prominence? 

*  Transpose  the  second  and  third  fractions. 


204  SCHOOL  ALGEBRA 

22.  From  what  language  does  the  word  algebra  come? 
What  does  the  word  algebra  mean? 

23.  Work  the  odd-numbered  examples  on  p.  101.  How 
many  examples  on  that  page  can  you  work  at  sight? 

128.  Two  Equivalent  Equations  are  equations  which  have 

identical  roots;  that  is,  each  equation  has  all  the  roots  of  the 

other  equation  and  no  other  roots. 

Thus,  X?  -  4:X  =  0,  and  x(x  -\- 2)  (x  -  2)  =0  are  equivalent, 
since  each  is  satisfied  by  the  values  x  =  0,  2,  —  2,  and  by  no  other 
values  of  x. 

If  we  multiply  the  two  members  of  an  equation  by  the 

same  expression,  the  resulting  members  are  equal,  but  the 

resulting  equation  may  not  be  equivalent  to  the  original 

equation. 

Thus,  if  we  take  the  equation  x  =  S  and  multiply  each  mem- 
ber by  a;  —  2,  we  obtain  x{x  —  2)  =  3 (x  —  2)  or 

{x  -S){x  -2)  =  0, 

which  is  not  equivalent  to  the  original  equation,  since  it  has  the 
root  X  =  2,  which  the  original  equation  does  not  have  (Art.  103). 

In  general,  if  the  two  members  of  an  integral  equation  are 
multiplied  by  x  —  a,  the  root  a  is  introduced  and  the  resulting 
equation  is  not  equivalent  to  the  original  equation. 

129.  An  Extraneous  Boot  is  a  root  introduced  into  an 
equation  (usually  unintentionally)  in  the  process  of  solving 
the  equation. 

The  simplest  way  in  which  an  extraneous  root  may  be 
introduced  is  by  multiplying  both  members  of  an  integral 
equation  by  an  expression  containing  the  unknown  number. 
See  the  example  of  Art.  128.  < 

A  more  common  way  in  which  extraneous  roots  are  intro- 
duced during  a  solution  —  and  one  more  difficult  to  detect-^ 


FRACTIONAL  AND  LITERAL  EQUATIONS  205 

I  is  by  a  failure  to  reduce  to  its  lowest  terms  a  fraction  con- 
tained in  the  original  equation. 

2a;  —  4 
Thus,  in  solving  z tyt ^  =1,  the  first  step  should  be  to 

2x—  A 

reduce  the  fraction  7 rr-? o^  *o  i^^  lowest  terms.  If  this  is  done, 

{X  —  L)  (X  —  z) 

2 

we  obtain  the  equation r  =  1,  whence  x=  S. 

X        i 

If,  however,  we  should  fail  to  reduce  the  fraction  to  its  lowest 
terms  and  should  multiply  both  members  by  (x  —  1)  (a;  —  2),  we 
obtain  2x  —  4:  =  x^  —  Sx  +  2,  whence  x^  —  dx  +  Q  =  0, 

or  {x  -  3)  (x-  2)  =  0,  anda:  =  2,  3. 

On  testing  both  of  these  results,  we  find  that  the  extraneous  root 
2  has  been  introduced. 

Often  the  fraction  which  can  be  reduced  to  simpler  terms 
occurs  in  a  disguised  and  scattered  form.  In  this  case  it  is 
best  to  solve  the  equation  without  attempting  to  collect  the 
parts  of  the  fraction.  An  extraneous  root  may  then  be 
detected  by  checking  the  results  obtained. 

Thus,  the  fraction  in  the  above  equation  might  be  changed 
in  the  following  way  so  as  to  make  it  difficult  to  detect  its 
presence  in  the  equation: 

We  have —^ — •  =  1 

{x  -l)(x  -  2)        ' 

2a;  —  2  2 

''^''"^  (»-l)(x-2)  -  (x-l)(x-2)  =  1' 

whence  '  _^_^__^_^=1. 

There  is  nothing  in  the  appearance  of  this  last  equation  to  indi- 
cate that  it  impUcitly  contains  a  fraction  which  should  be  simplified 
before  proceeding  with  the  solution  proper. 

Hence  it  is  important  constantly  to  remember  that  a  root  of  an 
equation  is  not  such  because  it  is  the  result  of  a  series  of  operations, 
as  clearing  an  equation  of  fractions,  transposition,  etc.,  but  because 
it  satisfies  the  original  equation. 


206  SCHOOL  ALGEBRA 

130.   Losing  Roots  in  the  Process  of  Solving  an  Equation. 

If  both  members  of  the  equation  (x  —  2)  {x  —  S)  =0  are  divided 
by  a:  —  2,  we  obtain  a;  —  3  =  0. 

The  resulting  equation  is  not  equivalent  to  the  original  equation 
since  it  does  not  contain  the  root  x  =  2,  which  the  original  equation 
contains. 

Hence,  in  general, 

//  both  members  of  an  equation  are  divided  by  an  expression 
containing  the  unknown  quantity,  write  the  divisor  expression 
equal  to  zero,  and  obtain  the  roots  of  the  equation  thu^  formed 
as  part  of  the  answer  for  the  original  equation. 

EXERCISE  62 

1.  Multiply  each  member  of  the  equation  a;  —  2  =  1  by 
X  —  2.  Is  the  resulting  equation  equivalent  to  the  original 
equation?    Why? 

2.  Make  up  and  work  an  example  similar  to  Ex.  1. 

3.  Multiply  each  member  of  the  equation  a;  =  2  by  x  —  5. 
Is  the  resulting  equation  equivalent  to  the  original  equation? 
Why? 

4.  Divide  each  member  of  the  equation  x^  —  9  =  x  —  3 
by  a:  —  3.  Is  the  resulting  equation  equivalent  to  the  orig- 
inal equation?    Why? 

5.  Make  up  and  work  an  example  similar  to  Ex.  4. 

X  —  S 

6.  Solve  the  equation  — — -  =  1  after  first  reducing  the 

fraction  to  its  lowest  terms.  Now  solve  the  equation  without 
reducing  the  fraction  to  its  lowest  terms.  Do  the  two  meth- 
ods of  solution  give  the  same  result?  Which  result  is  correct? 
Why? 

7.  Make  up  and  work  an  example  similar  to  Ex.  6. 
Solve  each  of  the  following,  check  each  result,  and  point  • 


I 


FRACTIONAL  AND  LITERAL  EQUATIONS  207 


out  each  extraneous  root,  giving  the  probable  reason  for  the 
occurence  of  such  a  root: 

8.    ?+       1  ^ 


2      a;+l      x+1 

9 


^'   x+1       ^      (a;+l)(a;-2) 

10. ; — 1 — ^+1=  5 


(x  +  2){x  +  3)   •  x  +  2 

11.     =  X 

0^2-1  6 

12.  ^^^    H ^ ^  =  3. 

a:2  -  1  ^  a;  -  1      a:  +  1 

13.  Form  an  equation  in  which  3  is  the  extraneous  root. 

14.  How  many  examples  in  Exercise  31  (p.  121)  can  you 
now  work  at  sight? 

131.  A  numerical  equation  is  an  equation  in  which  the 
known  quantities  are  expressed  by  figures.  Thus,  all  the 
equations  on  p.  199  are  numerical  equations. 

A  literal  equation  is  an  equation  in  which  some  or  all 
of  the  known  quantities  are  denoted  by  letters;  as  by  a,  by 
c  .  .  .,  or  m,  n,  p  .  .  . 

The  methods  used  in  solving  literal  equations  are  the  same 
as  those  used  in  solving  numerical  equations. 

Ex.    Solve  a{x  —  a)  =  h{x  —  b). 

ax  —  a^  =bx  —  b^ 
ax  —hx  =  a?  —h"^ 
{a  -  b)x  =a^  -¥ 

X  =  a  -]-b  Root 

Check.  a{x  —  a)  =  a{a  +b  —  a)  =  ah 

b{x  -b)  =b{a  +b  -b)  =ab 


208  SCHOOL  ALGEBRA 


EXERCISE  63 

Solve  for  x  and  check: 

1.   3a;  +  2a  =  a;  +  "^a. 

11. 

a^a;  =  (a  -  6)2  +  62a;. 

2.  9ax  -  36  =  2ax  +  46. 

12. 

(a—  6)a;  =  a2— (a  +  6)a;, 

3.  5ax  —  c  =  ax  —  5c, 

4.  ax  +  6  =  6a;  +  26. 

13. 

ax      hx  _a      h 
b        a       b      a 

5.   3ca;  =  a-(26-  a  +  ca;). 

14. 

a-\-  X        a  —  X 

6.   5a;  -  2aa;  =  3  -  6. 

a-2x      a-\-2x 

7.  2ax  -  3b  =  ex -\- 2d, 

8.  (x  +  a)(x-h)  =  x^. 

15. 

4a;  —  a      ^       x-\-  a 
2x  —  a             X  —  a 

9.   a6(a;  +  1)  =  a^  +  62a;. 
10.    (a;-l)(a;-2)  =  (a;-a)2. 

16. 

X    .X    .X         , 

-  +  7  +  -  =  a. 
a      6      c 

f(^»)=l(^«)• 


18 


^g    ax  —  b  bx  —  OCX  —  a_^ 

ab  be             ae 

20  ot^      ,  6a;     _  3a2a;  +  62 
3M^  3a  -  6  ~  9a2  -  62  * 

21  ^  —       ^  =       1       _  X 

a      a  —  b      a-\-b      b 

a^  —  X  62  —  a;      c2  —  a;      a2 


22. 


a 


„,    5a2-7a;  ,  a62  +  10a;      10c2  +  3a;  ,  5(a  -  c)   ,   62 

23.     . -1-  ^   —I -f-  — . 

3a6  5ae  66c  36  5c 


FRACTIONAL  AND  LITERAL  EQUATIONS  209 

a-\-  X      ^       X  X  —  1  X 


„,        a  a  ^^        a  a  —  1 

24.    = 25.    = 

X  —  a      1  a+1  a 

—  X 


a  X         a-\-  X 

26.  Make  up  and  solve  two  literal  equations. 

27.  How  many  examples  in  Exercise  35  (p.  131)  can  you 
now  work  at  sight? 

EXERCISE  64 

Oral 

Solve  the  following  orally,  without  transposing  any  term  contain- 
ing x: 

1.  4a;  =  —  12  ^.   ax  =b  7.   hx  +  c  =  d 

2.  Sx  =  a  5.    2x  -  4:  =  Q  8.   ax  +bx  =  c 

3.  ax  =  5  6.    ax  —  5  =  7  9.    ax  =  c  +  5x 


10.    6 

=  3a; 

11. 

10  = 

=  -5a; 

12. 

1  =  3. 

21. 

i=2 

X 

30. 

X      1 
3"4 

13. 

a  =  bx 

22. 

i-3 

31. 

*-i 

14. 

c  =  dx 

23. 

i-' 

32. 

--f 

15. 

3a;  =2 

.24. 

--i-' 

33. 

-1 

16. 

2x=  - 

4 
5 

25. 

'4'=-- 

34. 

bx 

a=  — 

c 

17. 

^-\ 

26. 

a 

35. 

X      3 
2  "5 

18. 

1 

27. 

|x=2 

36. 

1        4 

?=7 

19. 

h- 

28. 

h^ 

37. 

X           2 
4~       3 

20. 

l-^ 

-f> 

29. 

-lx=6 

38. 

1      1 

x~  2 

210  SCHOOL  ALGEBRA 


39. 

2      3 
x~5 

41. 

2      3 
S~  X 

40. 

7      1 
S~x 

42. 

2       1 
3^=5 

45. 

a+h 

P+q-    ^ 

47. 

46. 

'-J=p+q 

48. 

43. 

ax 

c 

b 

d 

4 

^?, 

44. 

X 

15 

m 


X  —  a 

a+  b  _ c+  d 
X  5 

49.   How  many  examples  in  Exercise  45  (p.  155)  can  you  now 
work  at  sight? 

EXERCISE  65 
Review 
Solve  for  x  and  check: 

_2 5 2_  X     x'^-l 

x-2     x+2~x^-4:'  2      a;-l~^- 

6x4-1     2x-  1     2a;- 4 
15  5  7a;- 16 

a;- 6       x+b     ^a^-b"^^ 


x-2a     x+2a      x^-4a^ 
g    3pa;-3ga;     p-2q  ^  P-Q_Q 
x^—  (f-         x-\-  q      q—  X 
x—2     X— 3_  X— 5     X— 6 
^*  a;-3~a;-4~a;-6     a;-7* 

a;—  5_  X 
~5"~2' 


7.   6+i(^-9-^)  +  3  + 


Find  the  value  of  x  in  the  shortest  way,  when 

8.  ^a;  =  ^X  19  +  ^X41. 

9.  3.1416a;  =  3.1416(723)  -  3.1416(476). 

,^    ^     ^     9  11    2a;     a:      - 

10.    3-^=2.  11.    3-2-5- 

12.   If  5  -  -  =  4,  find  the  value  of  x  when  ?/  =  ^'      Also  when 
X     y  o 

=  -  -•     When  2/  =  c* 
4  0 


FRACTIONAL  AND  LITERAL  EQUATIONS  211 

13.   Solve  for  I:  jt^  =    _  , 


r.     1  r  -640  f-l2S\ 

14.    Solve  for  a:  -^  =  a[-^Y) 


15.    In  adding  —r — h  -^  we  retain  the  L.  C.  D.  24.    In  solving 


or  ~~  1       ^x 
the  equation  —r—  =  -^  and  clearing  of  fractions,  the  L.  C.  D. 

24  disappears.    What  is  the  reason  for  this  difference? 

16.  Make  up  an  example  similar  to  Ex.  15. 

17.  Make  up  and  solve  an  equation  which  contains  fractions 
with  the  denominators  8,  2{x  —  1),  and  4. 

18.  Make  up  and  solve  an  equation  which  contains  fractions 
with  the  denominators  a  +  6,  a  —  6,  and  b^  —  a^. 

EXERCISE  66 

1.  Find  the  number  the  sum  of  whose  third,  fourth,  and 
fifth  parts  is  94. 

2.  Make  up  and  work  a  problem  concerning  one  fourth  and 
one  sixth  of  some  number. 

3.  State  ^  —  -  =  28  as  a  problem  concerning  a  number 

and  find  the  number. 

4.  A  certain  number  exceeds  the  sum  of  its  third,  fourth, 
and  tenth  parts  by  38.    Find  the  number. 

5.  A  piece  of  bronze  weighs  415  pounds.  It  contains  twice 
as  much  zinc  as  tin,  and  8  times  as  much  copper  as  tin.  How 
many  pounds  of  each  material  are  in  the  bronze? 

6.  Find  two  consecutive  numbers  such  that  one  seventh 
of  the  greater  exceeds  one  ninth  of  the  less  by  1. 

7.  Express  in  symbols  15%  of  x.    5%  of  x.    115%  of  b. 


212  SCHOOL  ALGEBRA 

8.  Two  men  kept  a  store  for  a  year  and  made  $4800. 
The  man  who  owned  the  store  building  received  40%  more 
of  the  profits  than  the  other.    How  much  did  each  receive? 

9.  In  building  a  macadam  road  the  county  pays  twice 
as  much  as  the  state,  and  the  township  pays  50%  more  than 
the  state.   How  much  does  each  pay  if  the  road  costs  $18,000? 

10.  Separate  $770  into  two  parts  so  that  one  shall  exceed 
the  other  by  20%.    By  33j%. 

11.  The  difference  of  two  numbers  is  9.  3  increased  by 
Yi  of  the  less  of  the  two  numbers  equals  f  of  the  greater. 
Find  the  numbers. 

12.  The  iron  ore  in  the  United  States  is  J  of  the  iron  ore 
in  the  rest  of  the  world.  If  there  are  75,000,000,000  tons  of 
iron  ore  in  the  entire  world,  how  many  tons  of  iron  ore  are 
there  in  the  United  States? 

13.  The  population  of  India  is  f  that  of  China,  and  the 
population  of  the  rest  of  the  world  is  3|  times  that  of  India. 
What  is  the  population  of  India  and  China,  if  that  of  the 
entire  world  is  1,500,000,000? 

14.  A  man  bequeathed  $60,000  to  his  wife  and  three  chil- 
dren. In  a  first  will  he  bequeathed  his  wife  three  times  as 
large  a  share  as  one  child  received.  Later  he  changed  his 
will  and  bequeathed  his  wife  $10,000  more  than  the  share  of 
a  child.  By  which  of  the  two  wills  would  she  have  received 
the  larger  amount? 

15.  In  one  kind  of  concrete,  the  parts  of  cement,  sand, 
and  gravel  are  1,  2,  and  4.  In  another  kind  of  concrete  these 
parts  are  1,  3,  5.  How  many  more  cubic  feet  of  cement  are 
needed  to  make  5600  cu.  ft.  of  concrete  of  the  first  kind  than 
of  the  second? 


FRACTIONAL  AND  LITERAL  EQUATIONS  213 

16.  A  girl's  grades  are,  arithmetic  87,  reading  92,  and 
j>c()graphy  85.  What  grade  must  she  have  in  spelUng  to 
make  her  general  average  90? 

17.  The  average  wheat  crop  of  the  United  States  for 
four  years  was  660  millions  of  bushels.  What  would  the 
crop  for  the  fifth  year  need  to  be  in  order  to  make  the 
average  for  the  five  years  700  million  bushels? 

18.  A  pupil  has  worked  15  problems.  If  he  should  work 
9  more  problems  and  get  8  of  them  right,  his  average  would  be 
.75.    How  many  problems  has  he  worked  correctly  thus  far? 

19.  A  baseball  nine  has  played  36  games  of  which  it  has 
won  25.  How  many  games  must  it  win  in  succession  to 
bring  its  average  of  gam-es  won  up  to  .75? 

20.  Make  up  and  work  an  example  similar  to  Ex.  19. 

21.  A  baseball  nine  has  won  19  games  out  of  36  games 
played.  If  after  this  it  should  win  f  of  the  games  played, 
how  many  games  would  it  need  to  play  to  bring  its  average 
of  games  won  up  to  .66f  ? 

22.  A  baseball  nine  has  won  25  games  out  of  36  played. 
It  still  has  12  games  to  play.  How  many  of  these  will  it  need 
to  win  in  order  to  bring  its  average  of  games  won  up  to  .75? 

23.  How  much  water  must  be  added  to  50  gallons  of 
milk  containing  8%  of  butter  fat  to  make  a  mixture  contain- 
ing 5%  of  butter  fat? 

SuG.     The  50  gal.  of  milk  contain  50  X  .08  or  4  gal.  butter  fat. 

4  5 

If  X  denotes  the  number  of  gallons  of  water,  — =  — — ,  etc. 

50  +  a;       100 

24.  A  certain  kind  of  cream  is  f  butter  fat,  and  a  certain 
kind  of  milk  is  3%  butter  fat.  How  many  gallons  of  the 
cream  must  be  added  to  40  gallons  of  milk  to  make  a  mixture 
which  is  5%  butter  fat? 


214  SCHOOL  ALGEBRA 

25.  Of  what  type  is  each  of  the  above  problems  an  example 
or  variation? 

26.  A  mass  of  copper  and  silver  alloy  weighs  120  lb.  and 
contains  8  lb.  of  copper.  How  much  copper  must  be  added 
to  the  mass  in  order  that  100  lb.  of  the  resulting  alloy  shall 
contain  10  lb.  of  the  copper? 

27.  A  mass  of  copper  and  silver  alloy  weighs  120  lb.  and  con- 
tains 8  lb.  of  silver.  How  much  silver  must  be  added  to  the 
mass  in  order  that  1  lb.  of  the  resulting  alloy  shall  contain 
2j  oz.  of  silver? 

28.  If  100  lb.  of  sea  water  contains  2|  lb.  of  salt,  how  much 
fresh  water  must  be  added  to  it  in  order  that  100  lb.  of  the 
mixture  shall  contain  1  lb.  of  salt? 

29.  How  much  fresh  water  must  be  added  to  100  lb.  of  sea 
water  in  order  that  20  lb.  of  the  mixture  shall  contain  4  oz. 
of  salt? 

30.  How  much  water  must  be  evaporated  from  100  lb. 
of  salt  water  in  order  that  8  lb.  of  the  water  left  shall  con- 
tain 1  lb.  of  salt? 

31.  How  much  water  must  be  added  to  a  gallon  of  alcohol 
which  is  90%  pure,  in  order  to  make  a  mixture  which  is  80% 
pure? 

32.  If  it  takes  a  man  9  days  to  do  a  piece  of  work,  what 
part  of  it  will  he  do  in  one  day?  If  it  takes  him  x  days  to 
do  the  work,  what  part  of  it  will  he  do  in  one  day? 

33.  If  a  boy  can  do  a  piece  of  work  in  15  days  which  a 
man  can  do  in  9  days,  how  long  would  it  take  both  working 
together  to  do  the  piece  of  work? 

SuG.  What  fractional  part  of  the  work  will  the  boy  do  in  1  day? 
The  man?  If  together  the  boy  and  man  can  do  the  piece  of  work 
in  X  days,  what  part  of  the  work  can  they  do  together  in  1  day? 


FRACTIONAL  AND  LITERAL  EQUATIONS  215 


B  34.   A  can  spade  a  garden  in  3  days,  B  in  4  days,  and  C  in 

'    o  days.    How  many  days  will  they  require  working  together? 

35.   A  and  B  together  can  mow  a  field  in  4  days,  but  A 

alone  could  do  it  in  12  days.    In  how  many  days  can  B  mow  it? 

|B  36.   A  and  B  in  5y  days  accomplish  a  piece  of  work  which 

I    A  and  C  can  do  in  6  days  or  B  and  C,  in  7j  days.    If  they  all 

work  together,  how  many  days  will  they  require  to  do  the 

same  work? 

37.  One  pipe  can  fill  a  given  tank  in  48  min.  and  another 
can  fill  it  in  1  h.  and  12  min.  How  long  will  it  take  the  pipes 
together  to  fill  the  tank? 

38.  Two  inflowing  pipes  can  fill  a  cistern  in  27  and  54 

min.  respectively,  and  an  outflowing  pipe  can  empty  it  in 

36  min.    All  pipes  are  open  and  the  cistern  is  empty;  in  how 

many  minutes  will  it  be  full? 

SuG.  Since  emptying  is  the  opposite  of  filling,  we  may  consider 
that  a  pipe  which  empties  jj  of  a  cistern  in  a  minute  will  fill  —  yV 
of  it  each  minute. 

39.  A  tank  has  four  pipes  attached,  two  filling  and  two 
emptying.  The  first  two  can  fill  it  in  40  and  64  min.  respect- 
ively, and  the  other  two  can  empty  it  in  48  and  72  min. 
respectively.  If  the  tank  is  empty  and  the  pipes  all  open,  in 
how  many  minutes  wifl  it  be  full? 

40.  At  what  time  between  3  and  4  o'clock  are  the  hands 
of  a  watch  pointing  in  opposite  directions? 

Solution.    At  3  o'clock  the  minute-hand  is  15  minute-spaces 
behind  the  hour-hand,  and  finally  is  30  spaces  in  advance:  therefore 
the  minute-hand  moves  over  45  spaces  more  than  the  hour-hand. 
Let  X  =  the  number  of  spaces  the  minute-hand  movea 

Then  X  -  45    =   "  "       "      "        "    hour-hand 

But  the  minute-hand  moves  12  times  as  fast  as  the  hour-hand; 

hence,  x  =  12(x  —  45).    Solving,  x  =  49xt. 
Thus  the  required  time  is  49  jj  min.  past  3. 


216  '         SCHOOL  ALGEBRA 

41.  When  are  the  hands  of  a  clock  pointing  in  opposite 
directions  between  4  and  5?    Between  1  and  2? 

42.  What  is  the  time  when  the  hands  of  a  clock  are  to- 
gether between  6  and  7?    Between  10  and  11? 

43.  At  what  instants  are  the  hands  of  a  watch  at  right 
angles  between  4  and  5  o'clock?    Between  7  and  8? 

44.  The  planet  Mars  is  in  the  most  favorable  position  to 
be  observed  from  the  earth  when  it  is  in 
line  with  the  earth  and  on  the  opposite 
side  of  the  earth  from  the  sun  (Mars 
is  then  said  to  be  in  opposition).  If 
the  year  is  taken  as  365  days,  and  it 
takes  Mars  687  days  to  make  one  revo- 
lution about  the  sun,  how  long  is  the 
interval  between  two  successive  opposi- 
tions of  Mars? 

SuG.  If  it  takes  the  earth  x  days  to  overtake  Mars  and  thus  put 
Mars  again  in  opposition,  how  many  revolutions  about  the  sun  does 
the  earth  make  in  x  days?  How  many  revolutions  does  Mars  make 
in  X  days?  In  the  interval  from  one  opposition  to  the  next,  how  many 
more  revolutions  about  the  sun  does  the  earth  make  than  Mars? 

45.  It  takes  the  planet  Jupiter  12  yr.  to  make  one  revolu- 
tion about  the  sun.  How  long  is  it  from  one  opposition  of 
Jupiter  to  the  next? 

46.  The  interval  between  two  successive  oppositions  of 
Mars  is  780  days.  Determine  the  time  it  takes  Mars  to  make 
one  revolution  about  the  sun  (i.  e.  the  length  of  the  year  on 
Mars). 

47.  A  courier  travels  5  mi.  an  hour  for  6  hours,  when  an- 
other courier  starts  at  the  same  place  and  follows  him  at  the 
rate  of  7  mi.  an  hour.  In  how  many  hours  will  the  second 
overtake  the  first? 


FRACTIONAL  AND  LITERAL  EQUATIONS         217 

SuG.  If  a;  =  the  number  of  hours  the  second  courier  travels, 
how  many  hours  does  the  first  courier  travel?  How  many  miles  (in 
terms  of  x)  does  the  first  courier  travel?  The  second?  Do  the  two 
couriers  travel  equal  distances? 

48.  A  courier  who  travels  5 J  mi.  an  hour  was  followed 
after  8  hours  by  another,  who  went  7 J  mi.  an  hour.  In  how 
many  hours  will  the  second  overtake  the  first? 

49.  A  woman  can  write  15  words  per  minute  with  a  pen, 
and  a  girl  can  write  40  words  per  minute  on  the  typewriter. 
The  woman  has  a  start  of  3  hours  in  copying  a  certain  manu- 
script. How  long  before  the  girl  using  the  typewriter  will 
overtake  the  woman? 

50.  A  train  running  40  mi.  an  hour  left  a  station  45  min. 
before  a  second  train  running  45  mi.  an  hour.  In  how  many 
hours  will  the  second  train  overtake  the  first? 

51.  A  gentleman  has  10  hours  at  his  disposal.  He  walks 
out  into  the  country  at  the  rate  of  3 J  mi.  an  hour  and  rides 
back  at  the  rate  of  4 J  mi.  an  hour.    How  far  may  he  go? 

52.  A  and  B  start  out  at  the  same  time  from  P  and  Q,  re- 
spectively, 82  mi.  apart.  A  walked  7  mi.  in  2  hours,  and  B 
10  mi.  in  3  hours.  How  far  and  how  long  did  each  walk 
before  coming  together,  if  they  walked  toward  each  other? 
If  A  walked  toward  Q,  and  B  in  the  same  direction  from  Q? 

53.  A  certain  room  is  20  ft.  long  and  12  ft.  wide.  The 
walls  and  ceiling  of  the  room  together  have  an  area  of  752 
sq.  ft.     How  high  is  the  ceiling? 

54.  A  rifle  ball  is  fired  at  a  target  1100  yd.  distant  and 
4j  sec.  after  firing  the  shot  the  marksman  heard  the  impact 
of  the  bullet  on  the  target.  If  the  bullet  traveled 
at  the  rate  of  2200  ft.  per  second,  what  was  the  rate 
at  which  the  sound  of  the  impact  traveled  back  to  the 
marksman? 


218  SCHOOL  ALGEBRA  j 

55.  A  rifle  ball  is  fired  at  a  target  1000  yd.  distant  and  4 
sec.  after  firing  the  shot,  the  marksman  heard  the  impact  of 
the  bullet  on  the  target.  If  sound  traveled  at  the  rate  of 
1100  ft.  per  second,  at  what  rate  did  the  bullet  travel? 

56.  A  21  lb.  mass  of  gold  and  silver  alloy  when  immersed 

in  water  weighed  only  19  lb.    If  the  gold  lost  yV  of  its  weight 

when  weighed  under  water,  and  the  silver  yV  of  its  weight, 

how  many  pounds  of  each  metal  were  in  the  alloy? 

SuG.  If  X  denotes  the  number  of  pounds  of  gold,  how  many 
pounds  of  silver  were  there  in  the  mass? 

The  law  involved  in  the  above  example  is  that  when  any  object 
is  weighed  in  water,  it  loses  in  weight  an  amount  equal  to  the  weight 
of  the  water  which  it  displaces.  Hence,  if  the  specific  gravity  of 
gold  is  approximately  19,  the  weight  of  the  water  displaced  by  the 
gold  =  yV  of  the  weight  of  the  gold. 

Find  out  if  you  can  who  first  used  this  method  of  determining 
the  relative  amounts  of  metal  in  an  alloy  and  what  use  he  first 
made  of  the  method. 

57.  An  alloy  of  aluminum  and  iron  weighs  80  lb.,  but 
when  immersed  in  water  it  weighs  only  60  lb.  If  the  spe- 
cific gravity  of  aluminum  is  2|  while  that  of  iron  is  7|,  how 
many  pounds  of  each  metal  are  in  the  alloy? 

58.  A  mass  of  copper  and  tin  weighing  100  lb.  when  im- 
mersed in  water  weighed  87.5  lb.  If  the  specific  gravity  of 
copper  is  8.8  and  that  of  tin  is  7.3,  how  much  of  each  metal 
was  in  the  mass? 

59.  If  a  bushel  of  oats  is  worth  40f^  and  a  bushel  of 
corn  is  worth  55^,  how  many  bushels  of  each  grain  must 
a  miller  use  to  produce  a  mixture  of  100  bu.  worth  4S^  a 
bushel? 

60.  A  man  has  $5050  invested,  some  at  4%,  and  some  at 
5%.  How  much  has  he  at  each  rate  if  the  annual  income  is 
$220? 


FRACTIONAL  AND  LITERAL  EQUATIONS  219 


increased  by  2,  the  second  diminished  by  2,  the  third  multi- 
plied by  2,  and  the  fourth  divided  by  2,  will  all  produce  equal 
results. 

62.  Find  three  consecutive  numbers  such  that  if  they  be 
divided  by  2,  3,  and  4  respectively,  the  sum  of  the  quotients 
will  equal  the  next  higher  consecutive  number. 

63.  In  the  United  States  the  gold  dollar  is  90%  gold  and 
10%  copper.  If  a  mass  of  gold  and  copper  weighing  24  lb.  is 
75%  gold,  how  many  pounds  of  gold  must  be  added  to  it  to 
make  it  ready  for  coinage  into  gold  dollars? 

64.  My  annual  income  is  $990.  If  J  of  my  property  is  in- 
vested at  5%,  I  at  4%,  and  the  rest  at  6%,  find  the  amount 
of  my  property. 

65.  If  one  pipe  can  fill  a  swimming  tank  in  1  hour  and  an- 
other can  fill  it  in  36  minutes,  how  long  will  it  take  the  two 
pipes  together  to  fill  the  tank? 

66.  At  what  time  are  the  hands  of  a  watch  at  right  angles 
between  10  and  11  o'clock? 

67.  If  one  baseball  nine  has  won  16  games  out  of  42 
played,  and  another  has  won  18  out  of  40  played,  how  many 
straight  games  must  the  first  team  win  in  order  at  least  to 
equal  the  average  of  games  won  by  the  second  team? 

68.  If  the  interval  between  two  successive  oppositions  of 
the  planet  Saturn  is  378  days,  how  long  is  the  year  on 
Saturn? 

69.  If  A,  B,  and  C  together  can  do  in  5^  days  a  certain 
amount  of  work,  which  B  alone  could  do  in  24  days,  or  C 
alone  in  16  days,  how  long  would  A  require? 

70.  How  much  water  must  be  added  to  1  gal.  of  a  5% 
solution  of  a  certain  chemical  to  reduce  it  to  a  2%  solution? 


220  SCHOOL  ALGEBRA 

71.  A  baseball  player  who  has  been  at  the  bat  150  times 
has  a  batting  average  of  .240.  How  many  more  times 
must  he  bat  in  order  to  bring  his  average  up  to  .250,  pro- 
vided that  in  the  future  his  base  hits  equal  half  the  number 
of  times  he  bats? 

72.  A  girl  has  worked  a  certain  number  of  problems  and 
has  f  of  them  right.  If  she  should  work  9  more  problems  and 
get  8  of  them  right,  her  average  would  be  .75.  How  many 
problems  has  she  worked? 

73.  If  the  sum  of  two  consecutive  integers  is  4p  +  5,  find 
the  integers. 

74.  A  man  has  a  hours  at  his  disposal.  He  wishes  to  ride 
out  into  the  country  and  walk  back.  How  far  may  he  ride 
in  a  coach  which  travels  b  miles  an  hour,  and  return  home  in 
time,  walking  c  miles  an  hour? 

75.  Generalize  Ex.  33;  that  is,  make  up  and  work  a  similar 
example  where  letters  are  used  instead  of  figures  for  the 
known  numbers. 

76.  If  E  denotes  the  number  of  days  it  takes  the  earth  to 
revolve  once  around  the  sun,  P  denotes  the  number  of  days 
it  takes  a  planet  (as  Mars)  to  complete  a  revolution  about 
the  sun,  and  S  the  number  of  days  between  two  successive 

oppositions  of  the  planet,  show  that  y?  —  is  =  o* 

77.  The  fore  wheel  of  a  carriage  is  a  feet  in  circumference 
and  the  hind  wheel  is  h  feet.  What  distance  has  been  passed 
over  when  the  fore  wheel  has  made  c  revolutions  more  than 
the  hind  wheel? 

78.  Make  up  and  work  three  examples  similar  to  such  of 
the  examples  in  this  Exercise  as  the  teacher  may  point  out. 


FRACTIONAL  AND  LITERAL  EQUATIONS  221 

EXERCISE   67 

1.  Given  V  =  Iwh,  find  h  in  terms  of  the  other  letters. 
Also  solve  for  I.    For  w, 

2.  Given  i  =  prt,  find  each  letter  in  terms  of  the  others. 
Find  each  letter  in  terms  of  the  others  in  the  following 

formulas  used  in  geometry: 

13.   K  =  ibh  6.  S  =  ttRL 

4.  Z  =  ihib  +  6')  7.  T  =  irRiR  +  L) 

5.   C  =  2'TrR  8.  r  =  2ttR{R  +  H) 

Also  find  each  letter  in  terms  of  the  others  in  the  following 
formulas  used  in  mechanics  and  physics: 

9.   S  -=  vt  11. 

10.  LW  =  Iw      12. 

15.  By  use  of  the  formula  in  Ex.  2  determine  in  how  many 
years  $325  will  produce  184.50  interest  at  5  per  cent. 

16.  Also  find  the  rate  at  which  $176  will  yield  $43.56  in- 
terest in  5  yr.  6  mo. 

17.  Change  the  following  temperatures  on  the  Centigrade 
scale  to  Fahrenheit  readings: 

(1)  50°  (2)  0°  (3)  2700° 

18.  Metals  fuse  at  the  following  temperatures  on  the  Cen- 
tigrade scale.  What  are  the  temperatures  at  which  they  fuse 
on  the  Fahrenheit  scale? 

Tin  228°       Lead  325°       Copper  1091°       Iron  1540° 

6c_-M 

19.  Solve  the  following  equation  for  h:       d       _  2d 
Also  solve  for  c.     For  d.  be  a 


C-I(F- 

-32) 

13.    R-     ^» 

9  +  s 

-i 

14.  i  =  i+L 

f    p    p' 

222  SCHOOL  ALGEBRA 

20.  A  boy  who  weighs  80  lb.  is  on  a  teeter  board  at  A, 

6  ft.  from  -the  fulcrum  F.     He  just  balances  a  boy  who 

^  jp         ^    is  at  ^  on  the  same  board,  8  ft. 

*^  Y^         jft — v^~~^     from  F.    What    does    the    second 

m  boy  weigh?     (Use  the  formula  of 

— * —         Ex.  10.) 

21.  Make  up  and  work  an  example  similar  to  Ex.  19. 

22.  How  many  examples  in  Exercise  48  (p.  163)  can  you 
now  work  at  sight? 


CHAPTER  XII 

SIMULTANEOUS  EQUATIONS 

132.  Need  and  Utility  of  Simultaneous  Equations. 

Ex.  A  farmer  one  year  made  a  profit  of  $2221  on  27 
acres  of  corn  and  40  acres  of  potatoes.  The  next  year  with 
equally  good  crops,  he  made  a  profit  of  $2028  on  36  acres  of 
corn  and  30  acres  of  potatoes.  How  much  did  he  make  per 
acre  on  his  corn  and  on  his  potatoes? 

Let        X  =  no.  of  dollars  made  on  1  acre  of  corn 

2/  =  "    "      ''  "      "  1    "    "  potatoes 

Then  27a;  +  40?/  =  2221 
36a;  +  30^  =  2028 

From  these  equations  the  value  of  x  may  be  found  by  combining 
the  equations  in  some  way  which  will  get  rid  of,  or  eliminate,  y. 
(See  Arts.  136-138.) 

Try  to  solve  the  above  problem  by  the  use  of  only  one  unknown, 
as  X.  If  you  succeed  at  all,  you  will  find  the  method  awkward  and 
inconvenient. 

Why  do  we  now  proceed  to  make  definitions  and  rules? 

133.  Simultaneous  Equations  are  a  set  or  system  of  equa- 
tions in  which  more  than  one  unknown  quantity  is  used,  and 
the  same  symbol  stands  for  the  same  unknown  number. 

I     Thus,  in  the  group  of  three  simultaneous  equations, 

a;  +  2/  +  20  =  13 
X  -2y  -\-z  ={) 
2x+y  -z  =S 

X  stands  for  the  same  unknown  number  in  all  of  the  three  equations, 
y  for  another  unknown  number,  and  z  for  still  another. 

223 


224  SCHOOL  ALGEBRA 

134.  Independent  Equations   are  those  which   cannot    be 

derived  one  from  the  other. 

Thus,  x-{-y  =  10, 

and  2x  =  20  -  2y, 

are  not  independent  equations,  since  by  transposing  2y  in  the  sec- 
ond equation  and  dividing  it  by  2,  we  may  convert  the  second  equa- 
tion into  the  first. 

But    Sx  —2y  =  5|   are  independent  equations,  since  neither  one 
5x  +y  =  Q]    of  them  can  be  converted  into  the  other. 

135.  Elimination  is  the  process  of  combining  two  equa- 
tions containing  two  unknown  quantities  so  as  to  form  a 
single  equation  with  only  one  unknown  quantity.  Or,  in 
general,  elimination  is  the  process  of  combining  several  sim- 
ultaneous equations  so  as  to  form  equations  one  less  in 
number  and  containing  one  less  unknown  quantity. 

There  are  three  principal  methods  of  elimination:  I,  ad- 
dition and  subtraction;  II,  substitution;  and  III,  comparison. 

These  methods  are  presented  to  best  advantage  in  connec- 
tion with  illustrative  examples. 

136.  I.  Elimination  by  Addition  and  Subtraction. 

Ex.    Solve  (9,_4  =  33 (3) 

In  order  to  make  the  coefficients  of  y  in  the  two  equations  ahke, 
we  multiply  equation  (1)  by  4,  and  (2)  by  5, 

48x  +  2%  =  300 (3) 

45a;  -  20?/  =  165 (4) 

Add  equations  (3)  and  (4),  93a;  =  465 
Divide  by  93,  a;  =  5  Root 

Substitute  for  x  its  value  5,  in  equation  (1), 
60  +  5?/  =  75 
.' .y  =3  Root 
Check.  12a;  +  5^  =  12  x  5  +  5  X  3  =75 

9a; -4?/  =9X5-4X3=  33 


H  SIMULTANEOUS  EQUATIONS  225 

Hpince  y  was  eliminated  by  adding  equations  (3)  and  (4) 
the  above  process  is  called  elimination  by  addition. 

The  same  example  might  have  been  solved  by  the  method 
of  subtraction. 

Thus,  multiply  equation  (1)  by  3,  and  (2)  by  4, 

36a;  +  15?/  =  225 (5) 

36a;  -  162/  =  132 (6) 

Subtract  (6)  from  (5),         31?/  =  93 

2/=3 
and      a;  =  5 


» 


It  is  important  to  select,  in  every  case,  the  smallest  multipliers 
that  will  cause  one  of  the  unknown  quantities  to  have  the  same 
coefficient  in  both  equations. 

Thus,  in  the  last  solution  given  above,  instead  of  multiplying 
equation  (1)  by  9,  and  (2)  by  12,  we  divide  these  multipUers 
by  their  common  factor,  3,  and  get  the  smaller  multipliers,  3 
and  4. 

Hence,  in  general, 

Multiply  the  given  equations  by  the  smallest  numbers  that 
mil  cau^e  one  of  the  unknown  quantities  to  have  the  same  co- 
efficient in  both  equations; 

If  the  equal  coefficients  have  the  same  sign,  subtract  the  corre- 
sponding members  of  the  two  equations;  if  the  equal  coefficients 
have  unlike  signs,  add, 

EXERCISE  68 

Solve  by  addition  and  subtraction: 

1.  Sx  -2y  =  1  4.   5a;  -  32/  =  1 

X  -{-  y  =  2  3a;  +  5y  =  21 

2.  2x  -  7y  =  9  5.     X  -\-  5y  ==  -  3 
5x-{-3y  =  2  7x-\-8y  =  6 

3.  4a;  +  3i/  =  ^1  6.   Sx  -  2y  =  4: 
2a; -62/ =  3  5a;  -  42/ =  7 


226  SCHOOL  ALGEBRA 


7.   2y  +  X  =  0 
4a:  +  61/  =  -  3 

13. 

3      5-^ 

8.     9a:  -  8y  =  5 

5  +  2~^ 

15a:  +  12y  =  2 
9.  4a:  -  6i/  H-  1  =  0 

14. 

3  ^4 

5a:  -  72/  +  1  =  0 
10.   8a:  +  5?/  =  6 

3a:      5i/  _ 

2^8 

dy  +  2x  =  11 

11.   5a:  -  3i/  =  36 
7a:  -  52/  =  56 

15. 

4^4.  32/ _       7 

5  "^  2 
3a:,  22/      7 

12.   ^-1  =  1 
2      3 

16. 

5a:      Sy 
6        9 

^      2/_  i 
4      9 

3a:      5?/ 
4        6 

17.  Find  two  numbers  whose  sum  is  12  and  whose  differ- 
ence is  2. 

18.  The  half  of  one  number  plus  the  third  of  another 
number  equals  13,  while  the  sum  of  the  numbers  is  33. 
Find  the  numbers. 

19.  State  Ex.  1  as  a  problem  concerning  two  numbers. 

20.  State  Ex.  2  as  a  problem  concerning  two  numbers. 

21.  7  lb.  of  sugar  and  3  lb.  of  rice  together  cost  57^;  also 
5  lb.  of  sugar  and  6  lb.  of  rice  cost  QO^.  Find  the  cost  of  a 
pound  of  each. 

22.  Make  up  and  work  an  example  similar  to  Ex.  18.  To 
Ex.  21. 

23.  How  many  examples  in  Exercise  50  (p.  170)  can  you 
now  work  at  sight? 


SIMULTANEOUS  EQUATIONS  227 

137.  II.  Elimination  by  Substitution. 

Ex.    Solve  5a;+22/  =  36  .    .    . (1) 

2a;  +  3!/  =  43 (2) 

From  (1)  5x  =  36  -  2?/ 

.-.      x  =  ^^ (3) 

In  equation  (2)  substitute  for  x  its  value  given  in  (3), 

^^+3.  =  43 

72 -4?/ +152/ =215 

11?/=  143 

2/  =  13  Root 

36—26 
Substitute  for  !/  in  (3),  x  =  — z =  2  Root 

Let  the  pupil  check  the  work. 
Hence,  in  general, 

In  one  of  the  given  equations  obtain  the  value  of  one  of  the 
unknown  quantities  in  terms  of  the  other  unknown  quantity; 
Substitute  this  value  in  the  other  equation  and' solve. 

EXERCISE  69 

1.  Work  the  examples  of  Exercise  68  (p.  225)  by  the 
method  of  substitution. 

Find  out  which  of  the  following  sets  of  equations  are  worked 
more  readily  by  the  method  of  addition  and  subtraction,  and 
which  by  the  method  of  substitution,  and  work  each  example 
accordingly: 

2.  X  =  Sy  -  5  4.     ir-3=0 
2x-\-5y  =  12  2y-\-Sx  =  5 

3.  Sx-  Ay  ^  1  5.   2x-{-Sy  =  1 
4x-  5y  =  1  3a;  +  4y  =  2 


228  SCHOOL  ALGEBRA 

e.  7x  +  8y  =  19  8.     y  =  3 

5x  +  62/  =  13J  2x  =  Sy  -  17 

7.     a;  =  2?/  —  3  9.  y  =  3x 

y  =  5x  -  21  4a;  +  5?/  =  38 

10.  Make  up  and  solve  an  example  in  simultaneous  equa- 
tions which  is  solved  more  readily  by  the  method  of  addition 
and  subtraction  than  by  the  method  of  substitution. 

11.  Make  up  and  solve  an  example  of  which  the  reverse 
of  Ex.  10  is  true. 

12.  How  many  examples  in  Exercise  51  (p.  172)  can  you 
now  work  at  sight? 

138.  III.  Elimination  by  Comparison. 

Ex.    Solve  2x-3y  =  23 (1) 

5x-^2y  =  29 (2) 

From(l)  2z=23+Sy (3) 

From  (2)  5x=29-22/ (4) 

From  (3)  x=?^ (5)^ 

From  (4)  x  =  ?^^ (6) 

Equate  the  two  values  of  x  in  (5)  and  (6),  //^^^"^ 

23+%..  29 -2y 

2       ""       5 

Hence,  115  +  15y  =  58  -  ^y 

19y  =  -  57 

2/  =  —  3  Root 

23—9 
Substitute  for  2/  in  (5),         x  =  — ^ —  =  ^  ^^^^ 

Let  the  pupil  check  the  solution. 
Hence,  in  general. 

Select  one  unknown  quantity,  and  find  its  valus  in  terms  of 
the  other  unknown  quantity  in  each  of  the  given  equations; 
Equate  these  two  values,  and  solve  the  resulting  equation. 


I 


SIMULTANEOUS  EQUATIONS  229 


EXERCISE  70 


1.  Work  the  examples  of  Exercise  68  (p.  225)  by  the 
method  of  comparison. 

Ascertain  by  which  of  the  three  methods  of  eUmination 
each  of  the  following  examples  can  be  worked  most  readily, 
and  solve  accordingly: 

2.  x  =  Sy-\-9  7.  9x  +  122/  =  -  6 
a;  =  5i/  +  13  6x  -    5y  =  -  17 

3.  a;  =  3i/  +  9  8.  a;  =  5 

Sx  -  5y  =  10  Sx  -  2y  =  13 

4.  6a;  +  5?/  -  8  =  0  9.   5a:  +  3i/  =  8 
4a;  -  32/  -  18  =  0  5x-4y  =  7 

5.  y  =  2x  10.   2/  =  f  (ic  -  3) 
3a;  +  22/  =  21  2/  =  f  a:  +  1 

6.  2/  =  6a;  -  3  11.  y  =  2x  +  1 
8-5x  =  y  3x  +  §  =  8 

12.  Make  up  and  solve  an  example  in  simultaneous 
equations  which  is  solved  more  readily  by  the  method  of 
comparison  than  by  either  of  the  other  two  methods  of  elim- 
ination. 

13.  Make  up  and  solve  an  example  in  simultaneous  equa- 
tions which  is  solved  more  readily  by  the  method  of  substi- 
tution than  by  either  of  the  other  two  methods. 

14.  Make  up  and  solve  an  example  solved  more  readily 
by  the  method  of  addition  and  subtraction  than  by  the  other 
two  methods. 


230  SCHOOL  ALGEBRA 

EXERCISE   71 

Solve  and  check  each  result: 

3         4  40  ^ 

1-3^  =  0  ~-'  +  ^y-h 

4y  _  ^+10  =  3  3y-x  =  2 


3 


2x  -  y  ,  Sx  +  2y  _„ 

2a;      4a:  +  y  -  1  _ 
3  "^  4 


6.  — — ^^—  =  0 

y  +  3      a;  +  4 

j/(a;  -  2)  -  a;(y  -  5)  +  13  =  0 

7.  f  (x  +  32/)  -  i{x  +  2y)  =  -jV 
32/-f(a;  +  4y  +  f)  =  0 

8.  .4a;  -  .32/  =  .7  10.  .5x  +  4.5t/  =  2.6 
.7a;  +  .2y  =  .5  1.3a;  +  3.1y  =  1.6 

9.  2a;  +  1.52/=  10  1^-   -Sa;  -  .7y  =  .005 
.3a;  -  my  =  .4  2a;  =  82/ 

12.       ^-3^^    =;r  +  13 
,  2-3a; 

IO2/+I  ^     _         2z  +  3 
5       ='     .  +  3-i±^ 


SIMULTANEOUS  EQUATIONS  '  231 

x-\-y     5 

y  _  3^      %  _  2^ 

3 2^_12 3^^2 

'         114  If 

14.    (x-5)(2/  +  3)  =  (a:-l)(2/  +  2) 
xy  +  2x=:  x(y  +  10)  +  72y 
x-2      x-\-10      lO-y_.^ 

2i/  +  6      4a:  +  i/  +  6      ^_Q 

3  8 

61/  +  5  3a:  +  54  _  %  -  4 

8  5a: -2^  12 

2y  +  3        x  +  y    _4y  +  7 

4  3x  -  22/  8 

3a: -2  ^  6a:  -  5  _  a:  +  y  +  64 
^^*        5              10  Qx-\-y 

3i/-2  ^  2?/ -5  3  + 7a: 

12      ~       8  lOy-Sx 

18.   Practice  oral  work  with  small  fractions  as  in  Exercise 
58  (p.  190). 

139.  Literal  Equations. 

Ex.    Solve  ax  -^by  =  c (1) 

ax  -\-l>y  =  c (2) 

Multiply  (1)  by  a!,  and  (2)  by  a, 

aa!x  +  a%y  =  dc (3) 

adx  +  ah'y  =  ac' (4) 

Subtract  (4)  from  (3),    {a!h  -  ah')y  =  a!c  -  ad 

dc  —  ad  ^    . 


232  SCHOOL  ALGEBRA 

Again,  multiply  (1)  by  6',  (2)  by  b, 

ah'x  +bh'y  =h'c (5) 

a'hx  +  hh'y  =  b(f (6) 

Subtract  (6)  from  (5),   (a6'  -  a'b)x  =  b'c  -  be' 

b'c  -  bc^  _    , 
ao  —  ao 
Let  the  pupil  check  the  work. 

In  solving  simultaneous  literal  equations,  observe  that  if  the 
value  obtained  for  the  first  unknown  is  a  fraction  containing  a 
binomial  term  (or  the  value  is  complex  in  other  ways),  it  is  better 
not  to  find  the  value  of  the  other  unknown  as  in  numerical  equations, 
i.  e.  by  substituting  the  value  found  in  one  of  the  original  equations 
and  reducing.  A  better  method  is  to  tak6  both  of  the  original  equa- 
tions and  eliminate  anew.   See  the  solution  of  the  preceding  example. 

EXERCISE  72 

Solve  and  check  each  result: 

1.  3x  +  4?/  =  2a  7.   ax  -\-hy  =  c 
5x-{-6y  =  4:a  mx  -\-  ny  =  d 

2.  2ax  +  Zby  =  ^ah  8.  6a:  +  ay  =  a  +  6 
5ax  4-  Ahy  =  Sab  ab{x  -  y)  =  a^  -  b^ 


3.   ax  -\-by  =  1 


9.   c^x  —  <Py  =  c  —  d 


ax  +  6V  -  1  ^^^2dx  -  cy)  =  2d''-c' 

4.  x  —  y  =  2n 

mx-ny  =  m'-\-  n^  lo.  ^  +  ^  =  i 

5.  26a;  +  a!/ =  46  + a  2/-^       ^ 
abx-2aby  =  46  +  a  x  +  y  =  2n 

6.  ax-by  =  a^-\-b''  H-    (a  +  l)x-by  =  a-\-2 
bx-\-ay  =  2(a2  +  6^)  {a  -  l)x  +  3by  =  9a 

12.    (a-6)a:- (a  +  6)2/  =  a2  +  62 
bx-{-  ay  =  0 

13.  -4-  +  -i^  =  2  15.  la7%+(a  +  %  ^^ 
a  +  6a-6  a^  +  b^ 

x-y  =  2b  ax-2by  =  a^  -  26^ 

14.  ax  —  hx  =  ay  —  dy  16.    (a  +  b)x  -\- cy  =  1 
x-y  =  l  CX+  {a-j-b)y  =  I 


SIMULTANEOUS  EQUATIONS  23L 

17.    (a  +  b)x  -  (a-  h)y  =  Sab 
(a  —  b)x  —  (a  +  b)y  =  ab 

a  +  o      a  —  0  0  —  1      0  —  a 

x-{-2a     y-2b_a^  +  b-  x-\-l      y-1      1 

a  —  b       a-\-b      a^—b^  b  1  —  a      b 

20.    (x  -l)(a-\-b)=a(y  +  a+  1) 
(y  +l){a-b)  =  b(x-b-  1) 

21.   Make  up  and  work  an  example  similar  to  Ex.  7.    To 
Ex.  11.  ^ 

140.  Three  or  More  Simultaneous  Equations. 

3a:  +  4!/  -  52  =  32 (1) 

Ex.    Solve  4a;  -  52/  +  3s  =  18 (2) 

5a:  -  3^  -  42  =  2 (3) 

If  we  choose  to  eliminate  z  first,  multiply  (1)  by  3,  and  (2)  by  5, 

9x  +  12y  -15z  =m (4) 

20a:  -^5y  +  15z  =90 (5) 

Add  (4)  and  (5),  29a:  -  13?/  =  186 (6) 

Also  multiply  (2)  by  4,  (3)  by  3, 

16a:  -  20?/  +  122  =  72 (7) 

15a;  -9y  -12z  =6      (8) 

Add  (7)  and  (8),  31a:  -  29?/  =  78 .  (9) 

We  now  have  the  pair  of  simultaneous  equations, 

1 29a;  -  13?/  =  186 
l31a:  -  29?/  =  78 
Solving  these,  obtain  a:  =  101    „ 

y  =8  \ 
Substitute  for  x  and  y  in  equation  (1), 

30  +  32  -  52  =  32, 

2=6    Root 

Check.  3a: +4?/- 52  =3x  10 +4x8-5x6=  32 

'4a: -5?/+ 32  =4x  10 -5X8+3X6  =  18 
5a: -3?/- 42  =5X  10 -3X8-4X6=2 


11. 


234  SCHOOL  ALGEBRA 

In  like  manner,  if  we  have  n  simultaneous  equations  con- 
taining n  unknown  quantities,  by  taking  different  pairs  of 
the. 71  equations,  we  may  eliminate  one  of  the  unknown  quan- 
tities, leaving  n  —  \  equations,  with  n  —  1  unknown  quanti- 
ties; and  so  on. 

EXERCISE   73 

Solve  and  check: 

1.  a:  +  2/  +  z  =  6  9. 
3a;  +  22/  +  z  =  10 
3a;  +  2/  +  33  =  14 

2.  3a;  -  2/  -  2z  =  11  10. 
4a;  -  2!/  +  2  =  -  2 
6a;  -  2/  +  33  =  -  3 

3.  5a;  -  62/  +  22  =  5 
8a;  +  42/  -  5z  =  5 
9a;  +  52/  —  6z  =  5 

4.  3a;-i2/  +  2  =  7| 
2x-\{y-?>z)  =  b\ 
2a;-i2/  +  42;  =  ll 

5.  2a;  +  32/  =  7  12.  a;  +  2/  +  22  =  2(a  +  h) 
32/  +  42  =  9  a;  4-  2  +  2i/  =  2(a  +  c) 
5a;  +  62  =  15  2/  +  2  +  2a;  =  2(6  +  c) 

6.  2a; +  42/ +  32  =  6  13.   a;  +  2/-2  =  3-a-6 
62/  -  3a;  +  22  =  7  a;  +  2-2/  =  3a-6-l 
3a;  -  82/  -  72  =  6  2/  +  2-a;  =  36-a-l 

7.  a;  +  32/  +  32  =  1  14.   3a;  +  22/  =  V-a 
3a;  -  52  =  1  62  -  2a;  =  f  6 

92/  +  IO2  +  3a;  =1  52/  -  132  +  a;  =  0 

Q.  u-\-x  —  w  =  4t  15-    — a;  +  2/  +  2  +  2)  =  a 
2z  +  2)  —  a;  =  1  a;  —  2/  +  2  +  2J  =  6 

2J  +  2(;  +  a;  =  8  a;  +  2/ —  2  +  2J  =  c 

2/ —  2(J  +  a;  =  5  a;  +  2/ +  2  —  2J  =  (Z 


ix 

+  h  +  h 

=  2 

ia; 

^\y  +  \z 

=  9 

1^ 

+  hy  +  ¥ 

=  3 

2a;  +  22/  -  2  = 

■2a 

3a; 

-y-Z  = 

46 

5a;  +  32/-32  = 

=  2(a  +  6) 

2a; 
3 

32/        42 

^4       5 

=  18 

5a; 
6 

52/32 

8  "^  4 

=  -5 

3a; 
2 

72/32. 

5       10 

=  -41 

I 


SIMULTANEOUS  EQUATIONS 


235 


Ex.  1.    Solve 


16.  Practice  the  oral  solution  of  simple  equations  as  in 
Exercise  64  (p.  209). 

141.  The  Use  of  ^  and  ;^  as  Unknown  Quantities  enables  us 
to  solve  certain  equations  which  would  otherwise  be  diffi- 
cult of  solution. 

^^  +  -  =  49 (1) 

X       y 

-  +  -  =  23      (2) 

KX      y 

Multiply  (1)  by  7,  and  (2)  by  5, 

(55+91  =  343 (3) 

5-^^=115 (4) 

x^  y  ^  ^ 

Subtract  (4)  from  (3),  —  =  228,  /.  -=  3,  or  ?/=  i  Root 

Substitute  the  value  of  y  in  (2),  hence,  x  =  \  Root 
Let  the  pupil  check  the  work. 


Ex.  2.    Solve 


I- +  #-=11 
2x      Zy 


(1) 

(2) 


X      4y       4 

When  X  and  y  in  the  denominators  have  coefficients,  as  in  this 
example,  it  is  usually  best  first  to  remove  these  coefficients  by  mul- 
tiplying each  equation  by  the  L.  C.  M.  of  the  coefficients  of  x  and  y 
in  the  denominators  of  that  equation.    Hence, 

Multiply  (1)  by  6,  and  (2)  by  4, 


x      y 

5-1=29 
X     y 


(3) 
(4) 


Solving  (3)  and  (4)  by  the  method  used  in  Ex.  1, 

'  X  =^] 

i\  Roots 

Let  the  pupil  check  the  work. 


236  SCHOOL  ALGEBRA 


EXERCISE 

74 

Solve  and  check; 

1.    ^  +  Z  =  3 

7. 

a      b  _a-h 

X      y 

X      y         a 

3  +  5  =  2 

h      a  _h  —  a 

X      y 

X      y          a 

.i-?=7 

8. 

X      y 

X       y 

3+-^  =  i 

1      1     ^                     .2 

-  +  -  =  m  +  ^ 

X      y 

X      y 

'■l^h=' 

9. 

ox      a?/ 

M-3 

X      y          ab 

*•  ^+1=1 

2x^iy 

10. 

«  +  6  +  «-6^2a 
ar              y 

^+3--5 

a;      y 

5.  A-A  =  iij 

11. 

by  -?,x  =  Ixy 

Ax      Zy      ''== 

15a:  +  my  =  IQxy 

M='«i 

12. 

W~+'~  =  2 
X      y      z 

6.1  +  1.1 

?-i+i=7 

X      y      n 

X      y      z 

1-1- 

3      2      5 

-  +  -  +  -  =  14 

X      y      '^ 

^ 

1 

a:      2/      2 

13.    -  - 

—  t 

^i 

X 

y 

5  , 

3 

-  + 

=  - 

-7 

y 

2 

2 

1 

' 

=  c 

1 

a: 

2 

SIMULTANEOUS  EQUATIONS  237 

2         1,4^1  ,1111 

14. =      3q77  15. =     - 

?  +  !-l  =  12  l_l_i  =  i 

X      y      2z  y      z      X      b 

5        3       1..  1111 


2x      4i/      3s  z      X      y      c 

X      y      z 

a  ,  c      b 

-H =  m 

X      z      y 

b   ,  c      a 

-H =  n 

y      z      X 

17.  5yz  +  6icz  —  3x1/  =  8a:i/z 
4:yz  —  ^xz  -\-  xy  =  l^xyz 
yz  —  12a;2  —  2xy  =  ^xyz 

18.  Make  up  and  work  an  example  similar  to  Ex.  1, 
To  Ex.  4.    Ex.  13.    Ex.  15. 

19.  Work  again  such  examples  on  pp.  212  and  213  as  the 
teacher  may  point  out. 

142.  In  the  Solution  of  Problems  Involving  Two  or  More 
Unknown  Quantities,  it  is  necessary  to  obtain  as  many  inde- 
pendent equations  as  there  are  unknown  quantities  involved 
in  the  equations  and  to  eliminate.    (See  Art.  134,  p.  224.) 

Ex.  Find  a  fraction  such  that  if  2  is  added  to  both  nu- 
merator and  denominator,  the  fraction  becomes  | ;  but  if  7  is 
added  to  both  numerator  and  denominator,  the  fraction  be- 
comes f . 

Two  unknown  numbers  occur  in  this  problem,  viz, :  the  numera- 
ator  and  denominator  of  the  required  fraction.  Hence  two 
equations  must  be  formed  in  order  to  obtain  a  solution  of  the 
problem. 


238  SCHOOL  ALGEBRA 

X 

Let  -  represent  the  fraction. 

Then,  ^^=2  ^^^  ^+T  =  3 

Clearing  these  equations,  and  collecting  like  terms, 

2x  -y  =  -  2 
Sx  -2y  =  -7 

The  solution  gives  x  =  S  and  ?/  =  8. 
Therefore  f  is  the  required  fraction. 
Let  the  pupil  check  the  work. 

EXERCISE    76 

1.  Find  two  numbers  whose  sum  is  23  and  whose  difference* 
is  5. 

2.  Twice  the  difference  of  two  numbers  is  6,  and  |  of  their 
sum  is  3|.    What  are  the  numbers? 

3.  Find  two  numbers  such  that  twice  the  greater  number 
exceeds  5  times  the  less  by  6;  but  the  sum  of  the  greater  num- 
ber and  twice  the  less  is  12. 

4.  2  lb.  of  flour  and  5  lb.  of  sugar  cost  31  cents,  and  5 
lb.  of  flour  and  3  lb.  of  sugar  cost  30  cents.  Find  the  value 
of  a  pound  of  each. 

5.  A  man  hired  4  men  and  3  boys  for  a  day  for  $18;  and 
for  another  day,  at  the  same  rate,  3  men  and  4  boys  for  $17. 
How  much  did  he  pay  each  man  and  each  boy  per  day? 

6.  In  an  orchard  of  100  trees,  the  apple  trees  are  5  more 
than  f  of  the  number  of  pear  trees.  How  many  trees  are 
there  of  each  kind? 

7.  One  woman  buys  4  yd.  of  silk  and  7  yd.  of  satin,  and 
another  woman  at  the  same  rate  buys  5  yd.  of  silk  and  5| 
yd.  of  satin.  Each  woman  pays  $17.70.  What  is  the  price 
of  a  yard  of  each  material? 


SIMULTANEOUS  EQUATIONS 


239 


8.  Solve  Ex.  7  without  using  x  and  y  to  represent  unknown 
numbers  (see  Art.  1).  About  how  much  of  the  labor  of  writ- 
ing out  the  solution  is  saved  by  the  use  of  x  and  i/? 

9.  1  cu.  ft.  of  iron  and  1  cu.  ft.  of  lead  together  weigh 
1180  lb.;  also  the  weight  of  3  cu.  ft.  of  iron  exceeds  the  weight 
of  2  cu.  ft.  of  lead  by  40  lb.  What  is  the  weight  of  1  cu.  ft. 
of  each  of  these  materials? 

10.  In  an  athletic  meet,  the  winning  team  had  a  score  of 
26  points  and  the  second  team  had  a  score  of  21  points.  If 
the  winning  team  took  first  place  in  7  events  and  second 
place  in  5  events,  while  the  second  team  took  6  firsts  and  3 
seconds,  how  many  points  does  a  first  place  count?  A  second 
place? 

11.  In  an  athletic  meet,  the  three  winning  teams  made 
scores  as  follows: 


Team 

Ists 

2ds 

3ds 

Total  Score 

A 
B 
C 

5 
3 

1 

2 
3 
4 

2 
1 
6 

33 
25 
23 

What  did  each  of  the  first  three  places  in  an  event  count  in 
this  meet? 

12.  Make  up  and  work  an  example  similar  to  Ex.  10. 

13.  Two  partners  agree  to  divide  their  profits  each  year  in 
such  a  way  that  one  partner  receives  $1000  more  than  f  of 
what  the  other  receives.  If  the  profits  for  a  given  year  are 
$10,000,  what  does  each  partner  receive? 

14.  Separate  240  into  two  parts  such  that  twice  the  larger 
part  exceeds  ^yg,  times  the  smaller  by  10. 


240  SCHOOL  ALGEBRA 

15.  If  the  cost  of  a  telegram  of  14  words  between  two 
cities  is  62^,  and  one  of  17  words  is  71^,  what  is  the  charge 
for  the  first  10  words  in  a  message  and  for  each  word  after 
that? 

16.  Make  up  and  work  an  example  similar  to  Ex.  15 
concerning  telegraph  rates  between  two  cities  near  yom: 
home. 

17.  A  farmer  one  year  made  a  profit  of  $1640  on  20  acres 
planted  with  wheat  and  30  acres  planted  with  potatoes. 
The  next  year,  with  equally  good  crops,  he  made  a  profit 
of  $1210  on  30  acres  planted  with  wheat  and  20  acres 
planted  with  potatoes.  How  much  per  acre  on  the  average 
did  he  make  on  each  crop? 

18.  In  three  successive  years,  the  farmer  raised  crops  with 
profits  as  follows: 

(1)  20  A.  wheat,  30  A.  corn,  40  A.  potatoes;  profits  $1720 

(2)  30  A.  wheat,  40  A.  corn,  20  A.  potatoes; "profits  $1520 

(3)  40  A.  wheat,  20  A.  corn,  30  A.  potatoes;  profits  $1440 
What  were  his  average  profits  per  acre  for  each  kind  of 

crop? 

19.  The  freight  charges  between  two  cities  on  400  lb.  of 
first-class  freight  and  600  lb.  of  second-class  freight  were 
$14.24,  while  the  charges  on  500  lb.  of  first-class  freight  and 
800  lb.  of  second-class  were  $18.48.  What  was  the  rate  per 
100  lb.  on  each  class? 

20.  The  freight  charges  on  shipments  between  two  places 
were  as  follows:  800  lb.  of  4th  class  +  500  lb.  of  5th  class  + 
700  lb.  of  6th  class,  $17.11;  1000  lb.  of  4th  class  +  600  lb.  of 
5th  class  +  800  lb.  of  6th  class,  $20.66;  600  lb.  of  4th  class  + 
1000  lb.  of  5th  class  +  900  lb.  of  6th  class,  $20.52.  Find  the 
rate  per  100  lb.  for  each  of  the  classes  named. 


SIMULTANEOUS  EQUATIONS  241 

21.  The  corn  and  wheat  crops  of  the  United  States  in  the 
year  1909  were  together  3,509,000,000  bu.;  the  corn  and  oat 
crops  3,779,000,000  bu.;  and  the  wheat  and  oat  crops, 
1,744,000,000  bu.  How  many  bushels  were  in  each 
crop?  • 

22.  One  cubic  foot  of  iron  and  one  cubic  foot  of  aluminum 
weigh  636  lb.;  a  cubic  foot  of  iron  and  one  of  copper  weigh 
1030  lb.;  a  cubic  foot  of  copper  and  one  of  aluminum  weigh 
706  lb.  How  much  does  one  cubic  foot  of  each  of  these  ma- 
terials weigh? 

23.  In  boring  holes  in  a  metal  plate,  three  circles  touching 
each  other  are  to  be  drawn,  the  distances 
between    their    centers    being    .865    in., 
,650    in.,    and    .790     in.,    respectively. 
Find   the   radius  of  each   of   the   three     J_ 
circles. 

24.  The  Eiffel  Tower  is  taller  than 
the  Metropolitan  Life  Building  of  New  York,  and  the  latter 
building  is  taller  than  the  Washington  Monument.  If  the 
difference  between  the  heights  of  the  first  two  is  284  ft.; 
between  the  first  and  last  is  429  ft.;  and  the  sum  of  the 
first  and  last  is  1539  ft.,  find  the  height  of  each. 

25.  A  ton  of  fertilizer  which  contains  60  lb.  of  nitrogen, 
100  lb.  of  potash,  and  150  lb.  of  phosphate  is  worth  $21.50; 
a  ton  containing  70,  80,  and  90  lb.  of  these  constituents  in 
order  is  worth  $19;  and  one  containing  80,  120,  150  lb.  of 
each  in  order  is  worth  $25.50;  what  is  the  value  of  one  pound 
of  each  of  the  constituents  named? 

26.  If  a  bushel  of  oats  is  worth  40^  and  a  bushel  of  corn  is 
worth  55f^,  how  many  bushels  of  each  must  a  miller  use  to 
produce  a  mixture  of  100  bu.  worth  48ff  a  bushel? 


242  SCHOOL  ALGEBRA 

27.  How  many  pounds  of  20^  coffee  and  how  many  pounds 
of  32^  coffee  must  be  mixed  together  to  make  60  lb.  worth 
28j;i  a  pound? 

28.  Make  up  and  work  an  example  similar  to  Ex.  27. 

• 

29.  If  two  grades  of  tea  worth  50}Z^  and  75f!f  a  pound  are  to 
be  mixed  together  to  make  100  lb.  which  can  be  sold  for  72^ 
at  a  profit  of  20%,  how  many  pounds  of  each  must  be  used? 

30.  A  farmer  wishes  to  combine  milk  containing  5%  of 
butter  fat  with  cream  containing  40%  of  butter  fat  in  order 
to  produce  20  gal.  of  cream  which  shall  contain  25%  of 
butter  fat.  How  many  gallons  of  milk  and  how  many  of 
cream  must  he  use? 

31.  A  man  has  $5050  invested,  part  at  4%,  and  the  rest 
at  5%.  How  much  has  he  invested  at  each  rate  if  his  annual 
income  is  $220? 

Can  you  work  this  example  by  use  of  one  unknown  quan- 
tity? 

32.  A  man  wishes  to  invest  part  of  $12,000  at  5%  and  the 
rest  at  4%  so  that  he  may  obtain  an  income  of  $500.  How 
much  must  he  invest  at  each  of  the  rates  named? 

33.  Make  up  and  work  an  example  similar  to  Ex.  32. 

34.  If  a  rectangle  were  3  in.  longer  and  1  in.  narrower  it 
would  contain  5  sq.  in.  more  than  it  does  now;  but  if  it  were 
2  in.  shorter  and  2  in.  wider  its  area  would  remain  unchanged. 
What  are  its  dimensions? 

SuG.  Draw  a  diagram  for  each  rectangle  considered  in  the  prob- 
lem.   See  Ex.  30,  p.  104. 

35.  If  a  rectangle  were  made  3  ft.  shorter  and  1|  ft.  wider, 
or  if  it  were  7  ft.  shorter  and  5^  ft.  wider,  its  area  would 
remain  unchanged.    What  are  its  dimensions? 


SIMULTANEOUS  EQUATIONS  243 

36.  A  party  of  boys  purchased  a  boat  and  upon  payment 

for  the  same  discovered  that  if  they  had  numbered  3  more, 

they  would  have  paid  a  dollar  apiece  less;  but  if  they  had 

numbered  2  less,  they  would  have  paid  a  dollar  apiece  more. 

How  many  boys  were  there,  and  what  did  the  boat  cost? 

SuG.  Let  X  =  the  number  of  boys,  and  y  =  the  number  of  dollars 
each  paid.     Then  xy  represents  the  number  of  dollars  the  boat  cost. 

37.  After  going  a  certain  distance  in  an  automobile,  a 
driver  found  that  if  he  had  gone  3  mi.  an  hour  faster,  he  would 
have  traveled  the  distance  in  1  hr.  less  time;  and  that  if  he 
had  gone  5  mi.  faster,  he  would  have  gone  the  distance  in 
1|  hr.  less.    What  was  the  distance? 

38.  Make  up  and  work  an  example  similar  to  Ex.  37. 

39.  If  a  baseball  nine  should  play  two  games  more  and 
win  both,  it  will  have  won  |  of  the  games  played.  If,  however, 
it  should  play  7  more  and  win  4  of  them,  it  will  then  have 
won  I  of  the  games  played.  How  many  games  has  it  so  far 
played  and  how  many  has  it  won? 

40.  If  a  physician  should  have  12  more  cases  of  diphtheria 
and  treat  10  of  them  successfully,  he  will  have  treated  f  of  his 
cases  successfully.  But  if  he  should  have  32  more  cases  and 
succeed  with  30  of  them,  he  will  have  succeeded  with  |  of  his 
cases.  How  many  cases  has  he  had  so  far  and  how  many  has 
he  treated  successfully  ? 

41.  If  1  be  added  to  the  numerator  of  a  certain  fraction, 
the  value  of  the  fraction  becomes  J;  but  if  1  be  subtracted 
from  its  denominator,  the  value  of  the  fraction  becomes  J. 
Find  the  fraction. 

42.  There  is  a  fraction  such  that  if  4  be  added  to  its  numer- 
ator the  fraction  will  equal  f ;  but  if  3  be  subtracted  from  its 
denominator  the  fraction  will  equal  f .    What  is  the  fraction? 


244  SCHOOL  ALGEBRA 

43.  Make  up  and  work  an  example  similar  to  Ex.  42. 

44.  A  certain  fraction  becomes  equal  to  ^  if  if  is  added  to 
both  numerator  and  denominator.  It  becomes  |  if  2J  is 
subtracted  from  both  numerator  and  denominator.  What  is 
the  fraction? 

45.  Find  two  fractions,  with  numerators  11  and  7,  respec- 
tively, such  that  their  sum  is  3y^,  but  when  their  denomina- 
tors are  interchanged,  their  sum  becomes  3^2  • 

46.  If  f  is  added  to  the  numerator  of  a  certain  fraction,  its 
value  is  increased  by  21 ;  hut  if  2^  is  taken  from  its  denomi- 
nator, the  fraction  becomes  f.    Find  the  fraction. 

47.  The  sum  of  two  numbers  is  97,  and  if  the  greater  is 
divided  by  the  less,  the  quotient  is  5  and  the  remainder  1. 
Find  the  numbers. 

SuG.  The  divisor  multiplied  by  the  quotient  is  equal  to  the  divi- 
dend diminished  by  the  remainder. 

48.  Divide  the  number  100  into  two  such  parts  that  the 
greater  part  will  contain  the  less  3  times  with  a  remainder 
of  16. 

49.  The  difference  between  two  numbers  is  40,  and  the 
less  is  contained  in  the  greater  3  times  with  a  remainder  of 
12.    Find  the  numbers. 

50.  Separate  50  into  two  such  parts  that  J  of  the  larger 
shall  exceed  |  of  the  smaller  by  2. 

51.  A  tank  can  be  filled  by  two  pipes  one  of  which  runs  4 
hr.  and  the  other  5;  or  by  the  same  two  pipes  if  the  first  runs 
3  hr.  and  the  other  8.  How  long  will  it  take  each  pipe  running 
separately  to  fill  the  tank? 

52.  Two  persons,  A  and  B,  can  perform  a  piece  of  work  in 
16  days.    They  work  together  for  4  days,  when  B  is  left 


SIMULTANEOUS  EQUATIONS  246 

alone,  and  completes  the  task  in  36  days.    In  what  time  could 
each  do  the  work  separately? 

53.  A  and  B  can  do  a  piece  of  work  in  8  da.;  A  and  C  can 
do  the  same  in  10  da.;  and  B  and  C  can  do  it  in  12  da.  How 
long  will  it  take  each  to  do  it  alone? 

54.  37  means  10  X  3  +  7.  Does  xy  mean  lOo;  +  y"i  Why 
this  difference? 

55.  How  would  you  write  a  number  whose  digits  in  order 
from  left  to  right  are  /,  m,  and  n?  Why  may  not  such  a 
number  be  expressed  as  Imnt 

56.  Express  in  symbols  a  number  whose  digits  in  order  are 
a,  6,  c,  and  d.    Whose  digits  are  Xy  y,  and  z.    x  and  y. 

57.  A  number  consists  of  two  digits  whose  sum  is  13,  and 
if  4  is  subtracted  from  double  the  number,  the  order  of  the 
digits  is  reversed.    Find  the  number.  ! 

58.  The  sum  of  the  digits  of  a  certain  number  of  two 
figures  is  5,  and  if  3  times  the  units'  digit  is  added  to  the 
number,  the  order  of  the  digits  will  be  reversed.  What  is 
the  number? 

59.  Twice  the  units'  digit  of  a  certain  number  is  2  greater 
than  the  tens'  digit;  and  the  number  is  4  more  than  6  times 
the  sum  of  its  digits.    Find  the  number. 

60.  In  a  number  of  3  figures,  the  first  and  last  of  which  are 
alike,  the  tens'  digit  is  one  more  than  twice  the  sum  of  the 
other  two,  and  if  the  number  is  divided  b'y  the  sum  of  its  digits, 
the  quotient  is  21  and  the  remainder  4.    Find  the  number. 

61.  An  oarsman  can  row  12  mi.  down  stream  in  2  hr.,  but 
it  takes  him  6  hr.  to  return  against  the  current.  What  is  his 
rate  in  still  water  and  what  is  the  rate  of  the  stream? 

Make  up  and  work  a  similar  example. 


246  SCHOOL  ALGEBRA 

62.  A  boatman  rows  20  mi.  down  a  river  and  back  in  8 
hr.;  he  can  row  5  mi.  down  the  river  while  he  rows  3  mi.  up 
the  river.    Find  the  rate  of  the  man  and  of  the  stream. 

63.  A  man  rows  down  a  stream  20  mi.  in  2|  hr.,  and  rows 
back  only  f  as  fast.  Find  the  rate  of  the  man  and  of  the 
stream. 

64.  3  cu.  ft.  of  cast  iron  and  5  cu.  ft.  of  wrought  iron  to- 
gether weigh  3750  lb.;  also  7  cu.  ft.  of  the  former  and  4  cu. 
ft.  of  the  latter  weigh  5070  lb.  What  is  the  weight  of  1  cu.  ft. 
of  each? 

65.  Regarding  the  orbits  of  the  earth  and  of  the  planet 
Mars  as  circles  whose  center  is  the  sun,  the  greatest  distance 
between  the  earth  and  Mars  at  any  time  is  234,000,000  mi., 
and  the  least  distance  between  them  is  48,000,000  mi.  How 
far  is  each  of  them  from  the  sun? 

66.  2  lb.  of  tea  and  5  lb.  of  coffee  cost  $2.50.  If  the  price 
of  tea  should  increase  10%  and  that  of  coffee  should  diminish 
10%,  the  cost  of  the  above  amounts  of  each  would  be  $2.45. 
Find  the  cost  of  a  pound  of  each. 

67.  Two  bins  contain  a  mixture  of  corn  and  oats,  the  one 
twice  as  much  corn  as  oats,  and  the  other  three  times  as 
much  oats  as  corn.  How  much  must  be  taken  from  each 
bin  to  fill  a  third  bin  holding  40  bu.,  to  be  half  oats  and  half 
corn? 

68.  If  A  gives  B  $10,  A  will  have  half  as  much  as  B;  but  if 
B  gives  A  $30,  B  will  have  |  as  much  as  A.  How  much  has 
each? 

69.  Two  grades  of  spices  worth  25fi  and  50^  a  pound  are 
to  be  mixed  together  to  make  200  lb.  which  can  be  sold  at 
52^  per  lb.  at  a  profit  of  30%.  How  many  pounds  of  each 
grade  must  be  used? 


SIMULTANEOUS  EQUATIONS  247 

70.  A  train  maintained  a  uniform  rate  for  a  certain  dis- 
mce.    If  this  rate  had  been  8  mi.  more  each  hour,  the  time 

)ccupied  would  have  been  2  hr.  less;  but  if  the  rate  had  been 
[0  mi.  an  hour  less,  the  time  would  have  been  4  hr.  more, 
^ind  the  distance. 

71.  If  the  greater  of  two  numbers  is  divided  by  the  less, 
the  quotient  is  3  and  the  remainder  3,  but  if  3  times  the 
greater  be  divided  by  4  times  the  less,  the  quotient  is  2  and 
the  remainder  20.    Find  the  numbers. 

72.  Why  are  we  able  to  solve  problems  like  Exs.  70  and 
71  by  algebra  and  not  by  arithmetic? 

73.  Find  two  numbers  whose  sum  is  a  and  whose  difference 
is  b. 

74.  If  a  pounds  of  sugar  and  b  pounds  of  coffee  together 
cost  c  cents,  while  d  pounds  of  sugar  and  e  pounds  of  coffee 
together  cost  /  cents,  what  is  the  price  of  one  pound  of  each? 

75.  If  a  bushel  of  oats  is  worth  p  cents,  and  a  bushel  of 
corn  is  worth  q  cents,  how  many  bushels  of  each  must  be 
mixed  to  make  r  bushels  worth  s  cents  per  bushel? 

76.  Find  a  fraction  such  that  if  a  be  added  to  both  nu- 
merator and  denominator  the  value  of  the  fraction  is  p/q; 
but  if  b  is  added  to  both  numerator  and  denominator,  the 
value  of  the  fraction  is  r/s. 

77.  Generalize  Ex.  34  (p.  242),  by  using  a  letter  for  each 
number  in  the  example. 

78.  Generalize  Ex.  53  (p.  245),  by  using  a  letter  for  each 
number  in  the  example. 

79.  Make  up  and  work  three  examples  similar  to  such  of 
the  examples  in  this  Exercise  as  you  think  are  most  interesting 
or  instructive. 


248  SCHOOL  ALGEBRA 

143.  ITtilities  in  Algebra. 

1.  Brevity  of  expressions  which  represent  numbers.    Brevity 

means  a  saving  of  time  and  energy. 

Thus,  for  instance,  for  "  number  of  feet  in  the  length  of  a  rect- 
angle," we  may  use  a  single  letter  as  x. 

2.  The  saving  of  space  also  opens  the  way  for  the  use  of 
auxiliary  quantity  of  various  kinds. 

See,  for  instance,  the  process  of  factoring  o^  +  a^"^  +  6^,  p.  153. 

3.  By  using  a  letter  to  represent  any  number  (of  a  given 
class),  we  are  able  to  discover  and  prove  general  laws  of  numbers. 

Thus,  (a  +by  =  a^  +  2ab  +  b^  is  true  for  any  numbers  whatever. 

As  an  example  of  the  discovery  of  new  and  useful  laws  of  number, 
we  may  take  the  case  where  we  know  half  the  sum  and  half  the  dif- 
ference of  two  numbers  and  desire  to  find  the  numbers  themselves. 
In  the  above  description  of  the  known  facts,  there  is  nothing  to 
suggest  a  method  of  obtaining  the  desired  end.    But  if  we  express 

the  given  facts  in  the  algebraic  form,  thus,      ^      and      ^    ,  it  is 

at  once  suggested  that  half  the  difference  added  to  half  the  sum  will 
give  a,  the  greater  of  the  two  numbers,  and  subtracted  will  give  b,  th(? 
smaller. 

It  may  be  well  to  notice  that  one  source  of  this  discovery  is  that 
in  the  algebraic  expression  we  used  separate  symbols,  a  and  b,  of 
nearly  equal  size  for  the  two  numbers  considered. 

4.  Combination  of  several  rules  into  one  formula. 

Thus,  the  single  formula  p  =  br  combines  three  cases  (and  rules) 
used  in  arithmetic  in  treating  percentage.  Similarly,  the  formula 
i  =  prt  covers  all  cases  used  in  treating  interest  in  arithmetic. 

This  advantage  comes  (1)  from  the  fact  that  a  letter  may  be 
used  to  represent  any  number.    See  3  above. 

(2)  From  the  fact  that  an  equation  can  be  solved  for  any  letter 
in  the  equation. 

(3)  From  the  approximately  uniform  size  of  the  letters  employed, 
which  suggests  that  we  treat  all  the  letters  alike  and  give  each  the 
leadership  in  turn. 

5.  The  use  of  letters  to  represent  unknown  numbers  often 
enables  us  to  begin  in  the  middle  of  a  complex  problem  and  work 


I 


SIMULTANEOUS  EQUATIONS  249 


in  several  directions  and  thus  solve  problems  which  otherwise 
we  could  not  analyze.    See  the  examples  on  pp.  242-243. 

6.  We  should  also  remember  constantly  that  the  symbols 
used  in  algebra  (and  the  advantages  coming  from  their  use) 
are  but  a  part,  or  detail,  of  the  more  general  subject  of  sym- 
bolism as  a  whole  and  of  its  utilities;  and  that  a  training  in 
algebra  should  give  a  better  grasp  of  the  whole  subject  of  symbols 
and  their  uses. 

EXERCISE   76 

1.  Abbreviate  the  following  as  much  as  you  can  by  use  of 
the  letter  x: 

I  a  certain  number  +  J  the  number  =  25.  How  much 
shorter  is  your  expression  than  the  given  expression? 

2.  Make  up  and  work  an  example  similar  to  Ex.  1. 

3.  Why  does  a  knowledge  of  algebra  suggest  to  us  that  a 
number  like  27001  can  be  factored  and  also  the  method  of 
doing  this,  while  a  knowledge  of  arithmetic  does  not  do  the 
same?    (See  Ex.  29,  p.  127.) 

4.  Is  a  railroad  ticket  a  symbol  or  representative  of  the 
money  paid  for  it?  What  are  the  advantages  in  the  use  of 
the  ticket?    The  disadvantages? 

5.  Discuss  in  the  same  way  a  check  drawn  on  a  bank  and 
used  in  paying  a  bill. 

6.  In  canceling  a  railroad  ticket,  what  are  the  advantages 
in  punching  the  ticket  as  compared  with  crossing  it  with  a 
pencil  mark?    With  burning  it? 

7.  What  is  a  newspaper  (or  a  book)  a  symbol  or  represen- 
tative of?  What  are  the  advantages  in  its  use?  The  disad* 
vantages? 


250  SCHOOL  ALGEBRA 

'  8.  A  certain  firm  occupied  a  building  running  from  10  to 
20  Barclay  St.  in  a  certain  city  as  their  place  of  business.  In 
advertising  in  one  magazine  they  gave  their  address  as  10 
Barclay  St. ;  in  another  they  gave  their  address  as  12  Barclay 
St.;  in  another  as  14  Barclay  St.  What  was  the  advantage 
in  doing  this?  By  this  means  what  double  use  was  made  of 
the  symbols  10,  12,  14,  etc. 

9.  If  a  teacher  has  a  set  of  papers  from  each  of  several 
classes,  what  is  the  advantage  in  arranging  them  at  different 
angles  when  piling  one  set  upon  another? 

10.  Can  you  give  another  instance  where  difference  of 
position  is  utilized  as  a  symbol? 

11.  What  are  the  advantages  in  using  a  flag  as  a  symbol 
or  representative  of  a  nation? 

12.  What  are  the  advantages  and  disadvantages  of  read- 
ing a  book  of  travels  as  compared  with  traveling? 

13.  Why  does  a  policeman  in  a  large  city  have  a  number 
as  well  as  a  name?  Name  other  classes  of  men  which  have 
numbers  as  well  as  names. 

14.  What  are  the  advantages  to  a  person  in  having  a  name? 

15.  Let  each  pupil  make  up  (or  collect)  and  work  as  many 
examples  as  possible  similar  to  the  examples  in  this  Exercise. 

SuG.  This  work  is  of  such  a  nature  that  it  may  readily  be  ex- 
tended in  various  directions  at  the  option  of  the  teacher. 


CHAPTER  XIII 

GRAPHS 

144.  A  variable  is  a  quantity  which  has  an  indefinite 
number  of  different  values. 

A  function  is  a  variable  which  depends  on  another  variable 

for  its  value. 

Thus,  the  area  of  a  circle  is  a  function  of  the  radius  of  the  circle; 
the  wages  which  a  laborer  receives  is  a  function  of  the  time  that  the 
man  works. 

A  graph  is  a  diagram  representing  the  relation  between  a 
function  and  the  variable  on  which  the  function  depends  for 
its  value. 

A  function  may  depend  for  its  value  on  more  than  one  variable; 
thus,  the  area  of  a  rectangle  depends  on  two  quantities  —  the  length 
of  the  rectangle  and  the  breadth.  The  present  treatment  of  graphs, 
however,  is  limited  to  functions  which  depend  on  a  single  variable. 

In  algebra  we  study  only  those  functions  which  have  a  definite 
value  for  each  definite  value  of  the  variable. 

145.  Uses  of  Graphs.  A  graph  is  useful  in  showing  at  a 
glance  the  place  where  the  function  represented  has  the 
greatest  or  least  value  and  where  it  is  changing  its  value  most 
rapidly,  and  in  making  clear  similar  properties  of  the  function. 

Graphs  of  algebraic  equations  are  useful  in  making  clear 
certain  properties  of  such  equations  which  are  otherwise 
difficult  to  understand.  A  graph  also  often  furnishes  a  rapid 
method  of  determining  the  root  (or  roots)  of  an  equation. 

251 


252 


SCHOOL  ALGEBRA 


^ 


146.  Framework  of  Reference.  Axes  are  two  straight 
lines  perpendicular  to  each  other  which  are  used  as  an  auxil- 
iary framework  in  constructing  graphs;  as  XX'  and  YY', 

The  X-axis,  or  axis  of  abscissas,  is  the  horizontal  axis;  as 
XX',     The  y-axis,  or  axis  of  ordinates,  is  the  vertical  axis; 

as  YY', 

The  origin  is  the  point 
in  which  the  axes  inter- 
sect; as  the  point  0, 

The  ordinate  of  a  point 
is  the  line  drawn  from 
the  point  parallel  to  the 
i/-axis  and  terminated  by 
the  ar-axis.  The  abscissa 
of  a  point  is  the  part  of 
the  ic-axis  intercepted  between  the  origin  and  the  foot  of  the 
ordinate.  Thus,  the  ordinate  of  the  point  P  is  AP,  and  the 
abscissa  is  OA. 

i7-    ;  +. 


The  ordinate  is 
sometimes  termed 
the  "2/"  of  a 
point,  and  the  ab- 
scissa, the  "  x  "  of 
a  point. 


(-3,2) 


X'\ — I — ^- 


i--+- 


B 
(-2,-2) 


Ordinates 
above  the  a:-axis 
are  taken  as  plus; 
those  below,  as 
minus.  Abscissas 
to  the  right  of 
the  origin  are  plus;  those  to  the  left  are  minus. 

The  co-ordinates  of  a  point  are  the  abscissa  and  the  ordi- 
nate taken   together.    They  are  usually  written   together 


J,0 


-» — I — I — I — I — I — \x 


r 


I 


GRAPHS  253 


in    parenthesis    with    the    abscissa    first    and    a    comma 
between. 

Thus,  the  point  (2,  4)  is  the  point  whose  abscissa  is  2  and  ordi- 
nate 4,  or  the  point  P  of  the  figure.  Similarly,  the  point  ( —  3,  2)  is 
Q;  (-2,  -2)  is  R;  and  (1,  -4)  is  S. 

The  quadrants  are  the  four  parts  into  which  the  axes  di- 
vide a  plane.  Thus,  the  points  P,  Q,  R,  and  S  lie  in  the  firsty 
second,  third  and  fourth  quadrants,  respectively. 

EXERCISE   77 

Draw  axes  and  locate  each  of  the  following  points: 

1.  (3,2),  (-1,3),  (-2,  -4),  (4,-1). 

2.  (2,§),  (-3, -IJ),  (5, -f),  (-2,1). 

3.  (2,  0)_,  (-3,  0),_(0,  4),  (0,  -1)^(0,  0). 

4.  (1,V2),  (l,-V2),(V3,0),(V5,-3),  (-iV5,2V2). 

5.  Construct  the  triangle  whose  vertices  are  (1,1),  (2,-2) 
(3,  2). 

6.  Construct  the  quadrilateral  whose  vertices  are  (2,  —  1), 
(-4,  -3),  (-3,  5),  (3,  4). 

7.  Plot  the  points  (0,  0),  (1,  0),  (2,  0),  (5,  0),  (-1,  0), 
(-3,  0). 

8.  Also  (0, 0),  (0, 1),  (0, 2),  (0, 3),  (0, 5),  (0,  -1),  (0,  -3). 

9.  All  points  on  the  x-axis  have  what  ordinate? 

10.  All  points  on  the  ?/-axis  have  what  abscissa? 

11.  Plot  the  following  pairs  of  points  and  find  the  distance 
between  each  pair  of  points: 

{1)  (6,  5),  (2,  8)  (3)  (3,  -6),  (-2,  6) 

(^)  (3,  0),  (0,  6)  U)  (0,  0),  (-3,  5) 

12.  Construct  the  rectangle  whose  vertices  are  (1,  3), 
re,  3),  (1,  -2),  (6,  -2),  and  find  its  area. 


254 


SCHOOL  ALGEBRA 


13.  Construct  the  rectangle  whose  vertices  are  (  —  3,  4), 
(4,  4),  (-3,  -2),  (4,  -2),  and  find  its  area. 

14.  Construct  the  triangle  whose  vertices  are  (-3,  -4), 
(-1,  3),  (2,  -4),  and  find  its  area. 

15.  In  which  quadrant  are  the  abscissa  and  ordinate  both 
plus?  Both  minus?  In  which  quadrant  is  the  aKscissa  minus 
and  the  ordinate  plus?  In  which  is  the  abscissa  plus  and  the 
ordinate  minus? 

16.  Practice  oral  work  with  small  fractions  as  in  Exercise 
58  (p.  190). 


Graphs  of  Equations  of  the  First  Degree 

147.  To  Construct  the  Graph  of  an  Equation  of  the  First 
Degree  Containing  Two  Unknown  Quantities,  as  x  and   y. 

Let  X  hive  a  series  of  convenient  values y  as  0,  1,  2,  3,  etc., 
-1,  -2,  -3,  etc.; 

r 


OJ 


GRAPHS 


255 


Find  the  corresponding  values  of  y; 

Locate  the  points  thus  determined,  and  draw  a  line  through 

these  points. 

Ex.    Construct  the  graph  of  the  equation  y  =  2x  —1. 

Construct  the  points  (0,  -1),  (1,  1),  (2,  3),  (3,  5), 
(  — 1,  —3),  (—2,  —5),  etc.,  and  draw  a  line  through 
them.  The  straight  Une  AB  is  thus  found  to  be  the 
graph  oi  y  =  2x  —  I. 


X 

y 

0 

-1 

1 

1 

2 

3 

3 

5 

etc. 

etc. 

-1 

-3 

-2 

-5 

etc. 

etc. 

148.  Linear  Equations.  It  will  always  be 
found  that  the  graph  of  an  equation  of  the 
first  degree  containing  not  more  than  two 
unknown  quantities  is  a  straight  line.    Hence, 

A  linear  equation  is  an  equation  of  the  first  degree. 

149.  Abbreviated  Method  of  Constructing  the  Graph  of 
a  Linear  Equation.  Since  a  straight  line  is  determined  by 
two  points,  in  order  to  construct  the  graph  of  an  equation 
of  the  first  degree  it  is  sufficient  to  construct  any  two  points  of 
the  graph  and  draw  a  straight  line  through  them. 

Ex.  1.    Graph  3y  -  2a;  =  6. 

When  a;  =0,?/  =  2; 
when  y=0,x=  —  3. 
Hence,  the  graph 
passes  through  the 
points  (0,  2)  and 
(-3,0),  or  CD  is  the 
required  graph. 

The  greater  the 
distance  between 
the  points  chosen,  the 
more  accurate  the 
construction  will  be. 
It  is  usually  advis- 
able to  test  the 
result    obtained    by 

locating  a  third  point  and  observing  whether  it  falls  upon  the 
graph  as  constructed. 


256  SCHOOL  ALGEBRA 

If  the  given  line  does  not  pass  through  the  origin,  or  near  the 
origin  on  both  axes,  it  is  often  convenient  to  construct  the  line 
',    by  determining  the  points  where  the  Une  crosses  the  axes. 

\      Ex.  2.    Graph  4x -}- 7y  =  1. 

\      When  X  =  0,y  =  j;  when  y  =  0,  x  =  I.     Hence,  the  graph 

'  passes  close  to  the  origin  on  both  axes.    Hence,  find  two  points  on 

the  required  graph  at  some  distance  from  each  other,  as  by  letting 

x  =  0,  9,  and  finding  y  =  |,  —5.    Let  the  pupil  construct  the  figure. 


EXERCISE   78 

Graph  the  following.  (It  is  an  advantage,  if  possible,  to 
draw  the  graph  line  in  red,  the  rest  of  the  figure  in  black  ink.) 

1.  y  =  x  +  2  7.   4x  -  5y  =  1 

2.y  =  x-2  8.  ?^  =  32^ 

3.  Sx  +  2y  =  6  ^ 

4.  3a:  -  22/  =  6  9'  ^  =  ^(2/  "  1) 

5.  3a:  -  5?/  +  15  =  0             \Q.  y  =  -  x 

6.  y  =  2x  11.  2/  =  4 

12.  If  a:  =  2,  show  that  whatever  value  y  has,  x  always 
=  2.    Hence  the  graph  of  a;  =  2  is  a  line  parallel  to  the  y-axis. 

13.  Graph  a:  =  0;  also  y  =  0. 

14.  Show  how  to  determine  from  an  inspection  of  a  linear 
equation  whether  its  graph  passes  through  the  origin;  near 
the  origin  on  one  axis;  near  the  origin  on  both  axes. 

15.  Graph  5x  -\-  Qy  =  1 ;  also  Qx  —  y  =  12. 

16.  Obtain  and  state  a  short  method  of  graphing  a  linear 
equation  in  which  the  term  which  does  not  contain  a:  or  y  is 
missing,  as  2y  —  3x  =  0. 


GRAPHS 


257 


Before  graphing  the  following,  determine  the  best  method 
of  constructing  each  graph,  and  then  graph: 

17.  x  +  2y  =  4:     20.   ja;  +  ^2/  =  2       23.   x  -  y  =  5 

18.  2y  =  X  21.   a:  =  -  3  24.   i/  +  2  =  0 

19.  5x-6y=l  22.   bx-\-4y  =  0      25.   3x-2y  +  -i=0 

26.  Construct  the  triangle  whose  sides  are  the  graphs  of 
the  equations,  y  —  2x  +  \=0,3y  —  x  —  7  =  0,y  +  3a:  +  11  =  0. 

27.  An  equation  of  the  form  y  =  b  represents  a  line  in 
what  position?    One  of  the  form  x  =  a? 

28.  Make  up  and  work  an  example  similar  to  Ex.  4.  To 
Ex.  26. 

150.  Graphic  Solution  of  Simultaneous  Linear  Equations. 
If  we  construct  the  graph  of  the  equation  x  —  y  =  3  (the 
line  AB)  and  the  graph  of  3x  -\-  2y  =  4:  (the  line  CD),  and 


c   -^ 


^  -4. 


258  SCHOOL  ALGEBRA 

measure  the  co-ordinates  of  their  points  of  intersection,  we 
find  this  point  to  be  (2,  —1). 

x-y  =  3 


If  we  solve  the  pair  of  simultaneous  equations  ,  ^  ,  ^  _  a 
by  the  ordinary  algebraic  method,  we  find  that  x  =  2  and 

y=-l. 

In  general,  the  roots  of  two  simultaneous  linear  equations  cor- 
respond to  the  co-ordinates  of  the  point  of  intersection  of  their 
graphs;  for  these  co-ordinates  are  the  only  ones  which  sat- 
isfy both  graphs,  and  their  values  are  also  the  only  values  of 
X  and  y  which  satisfy  both  equations. 

Hence,  to  obtain  the  graphic  solution  of  two  simultaneous 
equations. 

Draw  the  graphs  of  the  given  equations,  and  measure  the  co- 
ordinates of  the  point  {or  points)  of  intersection. 

Graphing  two  simultaneous  equations  forms  a  convenient 
method  of  testing  or  checking  their  algebraic  solution. 

151.  Simultaneous    Linear    Equations    whose    Graphs    are 

Parallel  Lines.     Construct  the  graph  of  x  -{-  2y  =  2  and  also 

of  3a: +  6?/  =  12. 

You  will  find  that  the  graphs  obtained  are  parallel  straight  lineis. 
Now  try  to  solve  the  same  equations  algebraically.  You  will  find 
that  when  either  a;  or  ?/  is  eliminated,  the  other  unknown  quantity 
is  eliminated  also,  and  that  it  is  therefore  impossible  to  obtain  a 
solution.  The  reason  why  an  algebraic  solution  is  impossible  is 
made  clear  by  the  fact  that  the  graphs,  being  parallel  fines,  cannot 
intersect;  that  is  to  say,  there  are  no  values  of  x  and  y  which 
will  satisfy  both  of  these  lines,  or  both  equations,  at  the  same 
time. 

152.  Graphic  Solution  of  an  Equation  of  the  First  Degree 
of  One  Unknown  Quantity.     By  substituting  for  y  in  the 

(y  —  X  —  3 
_  r.  the  two  equations 


GRAPHS  259 

reduce  to  a:  —  3  =  0.  Accordingly,  the  graphic  solution  of 
an  equation  like  x  —  3  =  0  can  be  obtained  by  combining 
the  graphs  oi  y  =  x  —  3  and  y  =  0.  In  other  words,  the  root 
of  a:  —  3  =  0  is  represented  graphically  by  the  abscissa  of 
the  point  where  the  graph  oi  y  =  x  —  3  crosses  the  a:-axis. 

EXERCISE   79 

Solve  each  pair  of  the  following  equations  both  graphically 
and  algebraically,  and  compare  the  results  in  each  example: 
^^    {2x-\-3y  =  7  ^     lx  +  7y  +  n=0 


2. 


x-y=^l  '    \x-3y  +  l=0 

y  =  3x-^  e     ly  =  ^ 

.y=-2x  +  l  '    \9x-5y  =  3 


^     \2y  =  x  1     \y^^ 

^'    \x  +  y-\-6  =  0  '    l2/  =  2a:  +  3 

(y  =  2x  ^'   Solve  graphically  2a:+3  =  0 

'    \x -{- y  =  0  9.   Solvegraphically  3a:— 5  =  0 

10.  Discover  and  state  the  relation  between  the  coefficients 
of  two  linear  simultaneous  equations  whose  graphs  are  par- 
allel lines.  , 

11.  Solve  graphically  |  g^  +  2^/  =  11.  ,  ^       ,    _  , 

^x  —  oy  —  o. 

12.  Solve  both  algebraically  and  graphically  \  n    _  g    _  r 

13.  Construct  the  quadrilateral  whose  sides  are  the 
graphs  of  the  equations,  x  —  2y  —  4:  =  0,  x  -\-  y  =  1, 
3y  —  5a;— 15  =  0,  a:  +  2?/  —  4  =  0,  and  find  the  co- 
ordinates of  the  vertices  of  the  quadrilateral. 

14.  Make  up  and  work  an  example  similar  to  Ex.  1.  To 
Ex.6. 

15.  How  many  examples  in  Exercise  26  (p.  110)  can  you 
now  work  at  sight? 


260 


SCHOOL  ALGEBRA 


163.  Graphic  Solution  of  Written  Problems. 

I.  Railway  Problems. 

Ex.  The  distance  between  New  York  and  Philadelphia 
is  90  mi.  At  a  given  time,  a  train  leaves  each  city,  bound 
for  the  other  city,  the  train  from  New  York  going  at  40  mi. 
an  hour  and  the  one  from  Philadelphia  at  30  mi.  In  how 
many  hours  will  they  meet,  and  at  what  distance  from  New 
York? 

The  train  dispatcher  represents  the  distance  between  the  stations 
by  the  Une  AB,  each  space  denoting  10  mi.    Each  space  on  AI  rep- 
/  resents   1  hour.    He  lo- 

cates E  three  units  to  the 
right  of  A  and  one  unit 
above  AB,  and  F  four 
units  to  the  left  of  B  and 
one  unit  above  AB.  He 
produces  AE  and  BF  to 
meet  at  C,  and  draws 
SS§s§8gg  (J  J)  perpendicular  to  AB. 

He  obtains  the  distance  from  A  at  which  the  trains  meet,  by 
measuring  AD  to  scale  (and  hence  determines  the  siding  at  which 
one  train  must  wait  for  the  other).  He  obtains  the  time  that  elapses 
before  the  trains  meet,  by  measuring  CD  to  scale. 

The  problem  may  also  be  solved  algebraically  in  the  same  way 
as  Exs.  57-61,  p.  87. 

The  advantage  of  the  graphical  method  is  that  in  this  solution 
it  is  easy  to  make  allowance  for  any  waits  which  trains  may  make 
at  stations.  Hence,  railroad  time-tables  are  often  constructed 
entirely  by  graphical  methods. 


4hr. 
Shr. 


2hr. 
Ihr. 


J 

^^ 

c:^ 

^ 

"f . 

M 

T*-^ 

^ 

.^ 

B 


II.   Problems  in  the  Mixture  of  Materials. 

Ex.  In  order  to  obtain  a  mixture  containing  20%  of  butter 
fat,  in  what  proportion  should  cream  containing  30%  of  fat 
be  mixed  with  milk  containing  4%? 


JJ 

20 

16 

4 

10 

GRAPHS  261 

Graphical  Solution 

We  construct  a  rectangle,  and  write  in  two  adjacent  corners 
(here  the  left-hand  corners)  the  per  cents 
of  fat  (30  and  4)  in  the  two  given  fluids; 
and  in  the  middle  of  the  rectangle  we 
write  the  per  cent  (20)  desired  in  the  mix- 
ture. The  differences  between  the  num- 
ber in  the  middle  and  the  numbers  in  the 
corners  (16  and  10)  are  then  found  and  placed  as  in  the  diagram. 
The  differences  thus  found  show  the  relative  amounts  of  the  given 
fluids  to  be  used;  viz. :  10  parts  of  milk,  and  16  of  cream. 

Now  solve  this  problem  algebraically  by  the  method  used  for 
Exs.  26-30,  pp.  241-242;  and  by  an  examination  of  this  solution, 
discover  for  yourself  the  reason  for  the  above  graphical  solution. 

EXERCISE   80 

Solve  the  following  problems  graphically: 

1.  The  distance  between  New  York  and  Philadelphia  is 
90  mi.  If  a  train  leaves  New  York  at  noon  and  goes  40  mi. 
an  hour,  and  another  train  leaves  Philadelphia  at  the  same 
time  and  travels  20  mi.  an  hour,  at  what  time  and  how  far 
from  New  York  will  they  meet? 

2.  Make  up  and  work  an  example  similar  to  Ex.  1. 

3.  The  distance  between  New  York  and  Buffalo  is  440 
mi.  If  a  train  leaves  New  York  at  11  a.  m.  and  travels  at 
the  rate  of  40  mi.  an  hour,  and  a  train  traveling  30  mi.  an 
hour  leaves  Buffalo  at  the  same  time,  at  what  time  and  how 
far  from  New  York  will  the  trains  meet? 

4.  Make  up  and  work  an  example  similar  to  Ex.  3. 

5.  In  order  to  obtain  a  mixture  containing  22%  butter 
fat,  in  what  proportion  must  cream  containing  32%  of  fat 
be  mixed  with  milk  containing  5%? 


262  SCHOOL  ALGEBRA 

6.  In  order  to  obtain  a  mixture  containing  28%  of  butter 
fat,  in  what  proportion  must  cream  containing  35%  of  fat 
be  mixed  with  cream  containing  2'b%t 

7.  Make  up  and  work  an  example  similar  to  Ex.  6. 

8.  In  what  proportion  must  oats  worth  50f!^  a  bushel  be 
mixed  with  corn  worth  80^  a  bushel  in  order  to  make  a  mix- 
ture worth  60^;^  a  bushel? 

9.  Make  up  and  work  a  similar  example  concerning  mixing 
different  grades  of  coffee. 

10.  The  distance  PQ  is  48  mi.  At  8  a.  m.  one  boy  starts 
from  P  and  walks  toward  Q  at  the  uniform  rate  of  4  mi.  an 
hour.  At  the  same  time  another  boy  starts  from  Q  on  a 
bicycle  and  rides  toward  P  at  the  rate  of  8  mi.  an  hour  but 
at  the  end  of  each  hour  of  riding  rests  \  an  hour.  By  means 
of  a  graph  determine  where  and  when  the  two  boys  will  meet. 

11.  Make  up  and  work  an  example  similar  to  Ex.  10. 

12.  How  many  examples  in  Exercise  27  (p.  112)  can  you 
now  work  at  sight? 

EXERCISE  81 

Review 

1.  Tell  the  degree  of  each  term  of 

5a;3  _  4^22^2  _  \\^  —  xy  +  xy^  —  xHj  +  Sa;^  —7y-\- 11. 

2.  Factor  (1)  x^  +  4. 

(2)  m?  —  2mn  +  n^  +  5m  —  5n. 

(3)  a2  -  n2  -  m2  -  2ab  +  2mn  +  ¥. 

3.  Factor  2(x^  -  1)  +  7{x^  -  1). 

Simplify: 

'  4    4x  —  5      4:  -\-x      2  _  x  —  5 
~45~~~30~'^3      ~18"* 
1  -  5a;      3a;  +  5      23;  -  3    ' 
6a;2-6"^4a;+4"^3  -3a;* 


p 


GRAPHS  263 


6. 

2 

r  ^ 

2         1 

1  -  a;      X 

(X- 

1)3'  {i-xr 

7 

m 

x^  - 

V^- 

^) 

.  (-^)' 

-<■*» 

9. 

1-- 
a 

2) 

/ 

a; 

-3' 

o 

~  a  +  x 

Solve: 

10. 

1^+: 

x-1      1 

^=     2    +3- 

5 

11. 

'+1 

-K-D- 

313     V 

a;+l\l 

2  /r 

12. 

3a;-! 

2     5a:+14 

3-2a;_ 

li. 

6 

7x+15 

4         "^ 

13. 

X 

a;4-l     X- 

8     a;-9 

x-2 

x—  1     a;- 

6     a;-7 

14. 

Sx 

2  ■ 

y. 

3" 

4 
'9* 

"■  '-i-^- 

10- a;     2/- 10 
3             4 

5x     Sy 

4"^  2 

=  3i. 

2?/+ 4 

2a;+?/     a;+13 

3 

8             4 

15. 

2y 

A 

-  x  = 
3^ 
a; 

=  4a:2/. 

17.    (a-b] 

(a;  +  (a +  6)?/=  a +6. 

y 

9. 

{x-y){a?-¥)  =  a^+b\ 

18.   .Sx+.2y=:  1.3. 

.32/ +.22  =.8. 

.32+. 2a;  =.9. 

19. 

X 

y' 

5 

'6* 

20.   Solve 

a'^b'ah' 

X- 

■y  = 

2 

15' 

X   ,y       1 

21.   Is  it  allowable  to  divide  each  term  in  16a;  =  96  by  16? 
Is  it  allowable  to  divide  each  term  of  16a;  —  96  by  16?    Give 
reasons. 


ax+1      r  ,       ,,      a{x^-  1)     ^, 

when  X 


264  SCHOOL  ALGEBRA 

22.  Obtain  the  value  of 
[a(x+l)- J, 

23.  Solve        1+^=1+^11^. 

24.  Why  is  it  proper  to  change  —x=—3  into  a;  =  3,  and  not 
proper  to  change  —x—3  into  a;  +  3? 

o         c 

25.  What  number  added  to  the  denominators  of  r  and  -j,  respec- 
tively, will  make  the  results  equal? 

26.  Does  r  equal  -— r?     Does  equal -+-?     Does 

a—b  a     0  c  c     c 

c  c      c 

— — T  equal  -+r?     Verify  your  statements  for  the  special  case 

when  a  =  4,  6  =  8,  and  c  =  2. 

27.  The  sums  of  three  numbers  taken  two  and  two  are  20,  29, 
and  27.    What  are  the  numbers? 

28.  Factor  8{x+  yY-  {2x-  yY. 

29.  What  is  the  advantage  of  being  able  to  add  the  same  number 
to  both  members  of  an  equation?  In  being  able  to  transpose  a  term? 
To  divide  both  members  of  an  equation  by  the  same  number?  (See 
Art.  70.) 

30.  Solve  (a  +  c)x -  (a-  c)y  =  2ab. 

(a  +  b)x  —  {a—  h)y  =  2ac. 

31.  Does  (a^  +  b^)  (a  +  b)  equal  a^  +  6^?  Verify  your  state- 
ment by  letting  a  and  b  have  convenient  numerical  values.  Can 
you  prove  your  statement  without  the  use  of  numbers? 

32.  If  K  =  ttR^  and  C  =  2nR,  find  K  when  C  =  10  and  tt  =  ^i^. 

33.  If  s  =  ^gt"^  and  v  =  gt,  find  s  when  g=  S2  and  v  =  64. 

34.  If  C  =  2nR  and  V  =  UR\  find  V  when  C  =  33  and  tt  =  ^^ 
(use  cancellation  wherever  possible). 

Solve: 

35.  4(x+2/)  +  ^-  =  13.         36.     3a;+--^  =  6. 

x-y  2x-Sy 

Si^-^y) —=  -1.  4x---?— =  25. 

x-y  2x  —  3y 

SuG.    Let  p  =  (x  -\-y),  q  = 


x-y 


37.   Show  that 
reduces  to 


GRAPHS  265 

12t{t^  +  2)2  (21'^  -  sy  -  6t^  (2t^  -  Sy  (t^  +  2) 

{t'+2y 

6(3^2+40  (2^2-3)=^ 
(f+2y 

38.  Graph   y=2x  +  b   when   6=1.      On  the   same   diagram 
graph  y=  2x+b  when  6=2.    When  6  =  —  1.    When  6=0. 

39.  Graph  y=ax+2  when  a=  1.      On   the   same   diagram 
graph  y=  ax+  2  when  a=  2,  3,  —1,  —3. 

40.  Graph  y=  3a;  +  2,  and  y=—ix+2on  the  same  diagram. 

41.  Make  up  and  work  an  example  similar  to  Ex.  38.    To  Ex.  39. 

42.  The  Fahrenheit  reading  at  the  boiling  point  of  alcohol  is 
95°  higher  than  the  Centigrade  reading.    Find  each  of  the  readings. 

43.  Make  up  an  example  similar  to  Ex.  42,  using  the  fact  that 
ether  boils  at  96°  Fahrenheit. 

44.  Give  the  value  of  -  -j-  a,    a  -v-  -,      —-x^-r-  -x^. 

a  a  5         5 

45.  What  is  the  reciprocal  of  -  +  t  ? 

a      0 

46.  Show  that  elimination  by  comparison  is  a  special  form  of 
elimination  by  substitution. 

47.  Show  that  eUmination  by  addition  and  subtraction  may  also 
be  regarded  as  a  form  of  elimination  by  substitution. 

48.  Eliminate  a  between  the  equations  F  =  Ma  and  s  =  ^af^. 

49.  Given  I  =  a+  {n  —  V)d  and  s=  ^(a+  I),  find  s  in  terms  of 

d,  n,  and  I. 

SuG.    What  letter  must  be  eliminated? 

rl—a 


50.   Eliminate  I  between  1=  ar"^  ^  and  s 


r-1 


CHAPTER   XIV 

HISTORY  OF  ELEMENTARY  ALGEBRA 

295.  Epochs  in  the  Development  of  Algebra.  Some  knowl- 
edge of  the  origin  and  development  of  the  symbols  and 
processes  of  algebra  is  important  to  a  thorough  under- 
standing of  the  subject. 

The  oldest  known  mathematical  writing  is  a  papyrus  roll, 
now  in  the  British  Museum,  entitled  *'  Directions  for  Attain- 
ing to  the  Knowledge  of  All  Dark  Things."  It  was  written 
by  a  scribe  named  Ahmes  at  least  as  early  as  1700  b.  c,  and 
is  a  copy,  the  writer  says,  of  a  more  ancient  work,  dating, 
say,  3000  b.  c,  or  several  centuries  before  the  time  of  Moses. 
This  papyrus  roll  contains,  among  other  things,  the  begin- 
nings of  algebra  as  a  science.  Taking  the  epoch  indicated 
by  this  work  as  the  first,  the  principal  epochs  in  the  develop- 
ment of  algebra  are  as  follows: 

1.  Egyptian :  3000  B.  C -1500  B.  C. 

2.  Greek  (at  Alexandria):  200  A.D.-400  A.D.  Principal 
writer,  Diophantus. 

3.  Hindoo  (in  India):  500  A.D.-1200  A.D. 

4.  Arab :  800  A.  D.-1200  A.  D. 

5.  European:  1200  A. D.-  Leonardo  of  Pisa,  an  Italian, 
pubhshed  in  1202  a.  d.  a  work  on  the  Arabic  arithmetic 
which  contained  also  an  account  of  the  science  of  algebra  as 
it  then  existed  among  the  Arabs.    From  Italy  the  knowledge 

266 


HISTORY  OF  ELEMENTARY  ALGEBRA  267 

of  algebra  spread  to  France,  Germany,  and  England,  where 
its  subsequent  development  took  place. 
We  will  consider  briefly  the  history  of 
I.  Algebraic  Symbols. 
II.  Ideas  of  Algebraic  Quantity. 
III.  Algebraic  Processes. 


I 


I.  History  of  Algebraic  Symbols 

296.  Symbol  for  the  Unknown  Quantity. 

1.  Egyptians  (1700  b.  c):  used  the  word  hau  (expressed, 
of  course,  in  hieroglyphics),  meaning  "  heap." 

2.  Diophantus  (Alexandria,  350  A.  d.?):  <5',  or  9°';  plural, 

99. 

3.  Hindoos  (500  a.  D.-1200  a.  d.)  :  Sanscrit  word  for 
"  color,"  or  first  letters  of  words  for  colors  (as  blue,  yellow, 
white,  etc.). 

4.  Arabs  (800  a.  D.-1200  a.  d.)  :  Arabic  word  for  "  thing  " 
or  '^  root  "  (the  term  root,  as  still  used  in  algebra,  originates 
here). 

5.  Italians  (1500  A.  D.) :  Radix,  R,  Rj. 

6.  BomboUi  (Italy,  1572  a.  d.):  kij 

7.  Stifel  (Germany,  1544):  yl,  B,  C, 

8.  Stevinus  (Holland,  1586) :  ® 

9.  Vieta  (France,  1591):  vowels  A,  E,  I,  0,  U. 
10.  Descartes  (France,  1637) :  x,  y,  z,  etc. 

297.  Symbols  for  Powers  (of  x  at  first) ;  Exponents. 

1.  Diophantus:  Bwafi^;,  or  8"  (for  square  of  the  unknown 
quantity);  kv^o<;,  or  /c"  (for  its  cube). 

2.  Hindoos:  initial  letters  of  Sanscrit  words  for  "  square  " 
and  "  cube." 


^(^^  SCHOOL  ALGEBRA 

3.  Italians  (1500  a.d.):  "census"  or  "  zensus  "  or  "s" 
(for  x'');  "  cubus  "  or  "  c  "  (for  a^). 

4.  Bombelli  (1579):  ®,  ®,  ®,  (for  x,  x\  :>?), 

5.  Stevinus  (1586):  ®,  (D,  ®,  (for  x,  x\  x?). 

6.  Vieta  (1591):  A,  A  qitadratus,  A  cubus  (for  x,  o^  3!^). 

7.  Harriot  (England,  1631) :  a,  aa,  aaa, 

8.  Herigone  (France,  1634) :  a,  a2,  a3. 

9.  Descartes  (France,  1637) :  x,  x'^,  a^. 

Wallis  (England,  1659)  first  justified  the  use  of  fractional 
and  negative  exponents,  though  the  use  of  fractional  expo- 
nents had  been  suggested  earlier  by  Oresme  (1350),  and  the 
use  of  negative  exponents  by  Choquet  (c.  1500). 

Newton  (England,  1676)  first  used  a  general  exponent,  as 
in  a*",  where  n  denotes  any  exponent,  integral  or  fractional, 
positive  or  negative. 

298.    Symbols  for  Known  Quantities. 

1.  Diophantus:  ixovahe^  (i.  e.  monads),  or  fx  . 

2.  Regiomontanus  (Germany,  1430) :  letters  of  the  alphabet. 

3.  Italians:  d,  from  dragma, 

4.  BombelH:®. 

5.  Stevinus:  ®. 

6.  Vieta:  consonants,  B,  C,  D,  F,  ,  ,  .". 

7.  Descartes:  a,  b,  c,  d. 

Descartes  possibly  used  the  last  letters  of  the  alphabet,  x,  tj,  z,  to 
denote  unknown  quantities  because  these  letters  are  less  used  and 
less  familiar  than  a,  b,  c,  d,  .  .  .  .  ,  which  he  accordingly  used  to  de- 
note known  numbers. 

239.    Addition  Sign.    The  following  symbols  were  used: 

1.  Egyptians:  pair  of  legs  walking  forward  {to  the  left),  -A. 

2.  Diophantus:  juxtaposition  (thus,  a6,  meant  a  -i-  b). 


I 


HISTORY  OF  ELEMENTARY  ALGEBRA  269 


3.  Hindoos:  juxtaposition  (survives  in  Arabic  arithmetic, 
as  in  2 1,  which  means  2+1). 

4.  Italians:  plus,  then  p  (or  e,  or  <j>), 

5.  Germans  (1489):  -h, +,  +. 

300.  Subtraction  Sign. 

1.  Egyptians:  pair  of  legs  walking  backward  (to  the  right), 
thus,  A_;  or  a  flight  of  arrows. 

2.  Diophantus:  ^  (Greek  letter  yjr  inverted). 

3.  Hindoos:  a  dot  over  the  subtracted  quantity  (thus,  mn 
meant  m  —  n). 

4.  Italians :  minus ,  then  M  or  m  or  de, 

5.  Germans  (1489) :  horizontal  dash,  — . 

The  signs  +  and  —  were  first  printed  in  Johann  Widman's 
Mercantile  Arithmetic  (1489).  These  signs  probably  originated  in 
German  warehouses,  where  they  were  used  to  indicate  excess  or 
deficiency  in  the  weight  of  bales  and  chests  of  goods.  Stifel  (1544) 
was  the  first  to  use  them  systematically  to  indicate  the  operations 
of  addition  and  subtraction. 

301.  Multiplication  Sign.  IMultiplication  at  first  was 
usually  expressed  in  general  language.    But 

1.  Hindoos  indicated  multiplication  by  the  syllable  hha, 
from  bharita,  meaning  "  product,"  written  after  the  factors. 

2.  Oughtred  and  Harriot  (England,  1631)  invented  the 
present  symbol,  X . 

3.  Descartes  (1637)  used  a  dot  between  the  factors  (thus, 
a-b). 

302.  Division  Sign. 

1.  Hindoos  indicated  division  by  placing  the  divisor  under 
the  dividend  (no  line  between).    Thus,  5  meant  c  -i-  d. 

2.  Arabs,  by  a  straight  line.    Thus,  a  —  6,  or  a  I  6,  or  — 

b 


270  SCHOOL  ALGEBRA 

3.  Italians  expressed  the  operation  in  general  language. 

4.  Oughtred,  by  a  dot  between  the  dividend  and  divisor, 

5.  Pell  (England,  1630),  by  -. 

303.  Equality  Sign. 

1.  Egyptians:  Z  □  (Also  other  more  complicated  sym- 
bols to  indicate  different  kinds  of  equality). 

2.  Diophantus:  general  language  or  the  symbol,  K 

3.  Hindoos:  by  placing  one  side  of  an  equation  immediately 
under  the  other  side. 

4.  Italians:  w  or  a;  that  is,  the  initial  letters  of  cequalis 
(equal).  This  symbol  was  afterward  modified  into  the  form, 
:» ,  and  was  much  used,  even  by  Descartes,  long  after  the 
invention  of  the  present  symbol  by  Recorde. 

5.  Recorde  (England,  1540):  =. 

He  says  that  he  selected  this  symbol  to  denote  equality  because 
"  than  two  equal  straight  Unes  no  two  things  can  be  more  equal." 

304.  Other  Symbols  used  in  Elementary  Algebra. 

Inequality  Signs,  >  <,  were  invented  by  Harriot  (1631). 

Oughtred,  at  the  same  time,  proposed  ~Z1,  j  as  signs  of  in- 
equality, but  those  suggested  by  Harriot  were  manifestly  superior. 

Parenthesis,  (  ),  was  invented  by  Girard  (1629). 

The  Vinculum  had  been  previously  suggested  by  Vieta 
(1591). 

Radical  Sign.  The  Hindoos  used  the  initial  syllable  of 
the  word  for  square  root,  Ka,  from  Karania,  to  indicate 
square  root. 

Rudolph  (Germany,  1525)  suggested  the  symbol  used  at 
present,  V,  (the  initial  letter,  r,  in  the  script  form,  of  the 
word  radix,  or  root)  to  indicate  square  root,  /W  to  denote 
the  4th  root,  and  /VW  to  denote  cube  root. 


r 


HISTORY  OF  ELEMENTARY  ALGEBRA  271 


Girard  (1633)  denoted  the  2d,  3d,  4th,  etc.,  roots,  as  at 
present,  by  v^,  V,  ^,  etc. 
The  sign  for  Infinity,   oo  ,  was  invented  by  WalHs  (1649). 

305.  Other  Algebraic  Symbols  have  been  invented  in  recent 
times,  but  these  do  not  belong  to  elementary  algebra. 

Other  kinds  of  algebra  have  also  been  invented,  employing 
other  systems  of  the  symbols. 

306.  General  Illustration  of  the  Evolution  of  Algebraic  Sym- 
bols. The  following  illustration  will  serve  to  show  the 
principal  steps  in  the  evolution  of  the  symbols  of  algebra: 

At  the  time  of  Diophantus  the  numbers  1,  2,  3,  4,  ...  .  were  de- 
noted by  letters  of  the  Greek  alphabet,  with  a  dash  over  the  letters 
used;  as,  a,~^,y,  .  .  .  . 

In  the  algebra  of  Diophantus  the  coefficient  occupies  the  last 
place  in  a  term  instead  of  the  first  as  at  present. 

Beginning    with    Diophantus,    the    algebraic    expression, 

x^  -{-  5x  —  4,  w^ould  be  expressed  in  symbols  as  follows: 

B^'a  ^oe^t^fiB  (Diophantus,  350  A.  D.) 

Iz  p.5  RmA  (Italy,  1500  a.  d.). 

IQ  +  5iV  -  4  (Germany,  1575). 

l^)p.5[i)mA[0]  (BombelH,  1579). 

1(2)  +  5®  -  4(0)       (Stevinus,  1586). 

1^^  +  5^-40       (Vieta,  1591). 

laa  -\-  5a    —  4  (Harriot,  1631). 

Ia2  +  5al  -  4  (Ilerigone,  1634). 

x'^  -}-  5x    —  4  (Descartes,  1637). 

307.  Three  Stages  in  the  Development  of  Algebraic  Sym- 
bols. 

1.  Algebra  without  Symbols  (called  Rhetorical  Algebra). 
In  this  primitive  stage,  algebraic  quantities  and  operations 
were  expressed  altogether  in  words,  without  the  use  of  sym- 


272  SCHOOL  ALGEBRA 

bols.   The  Egyptian  algebra  and  the  earUest  Hindoo,  Arabian, 
and  ItaUan  algebras  were  of  this  sort. 

2.  Algebra  in  which  the  Symbols  are  Abbreviated  Words 
(called  Syncopated  Algebra).  For  instance,  p  is  used  for  jdus. 
The  algebra  of  Diophantus  was  mainly  of  this  sort.  European 
algebra  did  not  get  beyond  this  stage  till  about  1600  a.  d. 

3.  Symbolic  Algebra.  In  its  final  or  completed  state, 
algebra  has  a  system  of  notation  or  symbols  of  its  own,  inde- 
pendent of  ordinary  language.  Its  operations  are  performed 
according  to  certain  laws  or  rules,  "  independent  of,  and  dis- 
tinct from,  the  laws  of  grammatical  construction.-" 

Thus,  to  express  addition  in  the  three  stages  we  have  plus, 
p,-\-;to  express  subtraction,  minus,  m,  — ;  to  express  equality, 
cequalis,  ce,  =, 

Along  with  the  development  of  algebraic  symbolism,  there 
was  a  corresponding  development  of  ideas  of  algebraic  quan- 
tity and  of  algebraic  processes. 

XL    History  of  Algebraic  Quantity 

308.  The  Kinds  of  Quantity  considered  in  algebra  are 
positive  and  negative;  particular  (or  numerical)  and  general; 
integral  and  fractional;  rational  and  irrational;  commensur- 
able and  incommensurable;  constant  and  variable;  real  and 
imaginary. 

309.  Ahmes  (1700  b.  c.)  in  his  treatise  uses  particular,  pos- 
itive  quantity,  both  integral  and  fractional  (his  fractions,  how- 
ever, are  usually  limited  to  those  which  have  a  unity  for  a 
numerator).  That  is,  his  algebra  treats  of  quantities  like  8 
and  I,  but  not  like  —  3,  or  —  f ,  or  V2,  or  —  a. 

310.  Diophantus  (350  a.  d.)  used  negative  quantity,  but 
only  in  a  limited  way;  that  is,  in  connection  with  a  larger 


HISTORY  OF  ELEMENTARY  ALGEBRA  273 

positive  quantity.  Thus,  he  used  7  —  5,  but  not  5  —  7,  or 
—  2.  He  did  not  use,  nor  apparently  conceive  of,  negative 
quantity  having  an  independent  existence. 

3U.  The  Hindoos  (500  A.  D.-1200  A.  D.)  had  a  distinct 
idea  of  independent  or  absolute  negative  quantity,  and  used 
the  minus  sign  both  as  a  quaUty  sign  and  a  sign  of  operation. 
They  explained  independent  negative  quantity  much  as  it  is 
explained  to-day  by  the  illustration  of  debts  as  compared 
with  assets,  and  by  the  opposition  in  direction  of  two  lines. 

Pythagoras  (Greece,  520  b.  c.)  discovered  irrational  quan- 
tity, but  the  Hindoos  were  the  first  to  use  this  in  algebra. 

312.  The  Arabs  avoided  the  use  of  negative  quantity  as 
far  as  possible.  This  led  them  to  make  much  use  of  the  pro- 
cess of  transposition  in  order  to  get  rid  of  negative  terms  in 
an  equation.  Their  name  for  algebra  was  "  al  gebr  we'l 
mukabala,"  which  means  "  transposition  and  reduction.** 

The  Arabs  used  surd  quantities  freely. 

313.  In  Europe  the  free  use  of  absolute  negative  quantity 
was  restored. 

Vieta  (1591)  was  principally  instrumental  in  bringing  into 
use  general  algebraic  quantity  (known  quantities  denoted  by 
letters  and  not  figures). 

Cardan  (Italy,  1545)  first  discussed  imaginary  quantities, 
which  he  termed  "  sophistic  *'  quantities. 

Euler  (Germany,  1707-83)  and  Gauss  (Germany,  1777- 
1855)  first  put  the  use  of  imaginary  quantities  on  a  scientific 
basis.    The  symbol  i  for  V  —  1  was  suggested  by  Gauss. 

Descartes  (1637)  introduced  the  systematic  use  of  variable 
quantity  as  distinguished  from  constant  quantity. 


274  SCHOOL  ALGEBRA 

III.    History  of  Algebraic  Processes 

314.  Solution  of  Equations.  Ahmes  solved  many  simple 
equations  of  the  first  degree,  of  which  the  following  is  an  ex- 
ample: 

"  Heap  its  seventh,  its  whole  equals  nineteen.   Find  heap." 

In  modern  symbols  this  is, 

X 

Given  --\-x  =  19;  find  x. 

The  correct  answer,  16f ,  was  given  by  Ahmes. 
Hero  (Alexandria,  120  b.  c.)  solved  what  is  in  effect  the 
quadratic  equatioriy 

where  d  is  unknown,  and  s  is  known. 

Diophantus  solved  simple  equations  of  one  unknown  quan- 
tity, and  simultaneous  equations  of  two  and  three  unknown 
quantities.  He  solved  quadratic  equations  much  as  is  done 
at  present,  completing  the  square  by  the  method  given  in 
Art.  226.  However,  in  order  to  avoid  the  use  of  negative 
quantity  as  far  as  possible,  he  made  three  classes  of  quadratic 
equations,  thus, 

ax^  -\-hx  =  c, 

ax^  -\-  c    =  hx, 

ax^  =  hx  -\-  c. 

In  solving  quadratic  equations,  he  rejected  negative  and 
irrational  answers. 

He  also  solved  equations  of  the  form  ax"^  =  6.r". 

He  was  the  first  to  investigate  indeterminate  equations,  and 
solved  many  such  equations  of  the  first  degree  with  two  or 
three  unknown  quantities,  and  some  of  the  second  degree. 


HISTORY  OF  ELEMENTARY  ALGEBRA  275 

The  Hindoos  first  invented  a  general  method  of  solving  a 
quadratic  equation  (now  known  as  the  Hindoo  method,  see 
Art.  233).  They  also  solved  particular  cases  of  higher  de- 
grees, and  gave  a  general  method  of  solving  indeterminate 
equations  of  the  first  degree. 

The  Arabs  took  a  step  backward,  for,  in  order  to  avoid  the 
use  of  negative  terms,  they  made  six  cases  of  quadratic  equar 
tions;  viz.: 

ao^  =  bXf  ax^  -\-bx  =  c, 

ax^  =  c,  ax^  +  c    =  bx, 

bx   =  c,  ax^  =  bx  -\-  c. 

Accordingly,  they  had  no  general  method  of  solving  a  quad- 
ratic equation. 

The  Arabs,  however,  solved  equations  of  the  form  ax'^^-i-bx^ 
=  c,  and  obtained  a  geometrical  solution  of  cubic  equations 
of  the  form  x^  +  pa:  +  g  =  0. 

In  Italy,  Tartaglia  (1500-1559)  discovered  the  general  so- 
lution  of  the  cubic  equation,  now  known  as  Cardan's  solution. 
Ferrari,  a  pupil  of  Cardan,  discovered  the  solution  of  eqvxir 
tions  of  the  fourth  degree. 

Vieta  discovered  many  of  the  elementary  properties  of  an 
equation  of  any  degree;  as,  for  instance,  that  the  number  of 
the  roots  of  an  equation  equals  the  degree  of  the  equation. 

315.  Other  Processes.  Methods  for  the  addition,  sub- 
traction, and  multiplication  of  polynomial  expressions  were 
given  by  Diophantus. 

Transposition  was  first  used  by  Diophantus,  though,  as  a 
process,  it  was  first  brought  into  prominence  by  the  Arabs. 
The  word  algebra  is  an  Arabic  word  and  means  "  transposi- 
tion "  {al  meaning  "  the,"  and  gebr  meaning  "  transposition"). 

The  Greeks  and  Romans  had  a  very  limited  knowledge  of 


276  SCHOOL  ALGEBRA 

fractions.  The  Hindoos  seem  to  have  been  the  first  to  reduce 
fractions  to  a  common  denominator. 

The  square  and  cube  root  of  polynomial  expressions  were 
extracted  by  the  Hindoos. 

The  methods  for  using  radicals,  including  the  extraction 
of  the  square  root  of  binomial  surds  and  the  rationalizing  of 
the  denominators  of  fractions,  were  also  invented  by  the 
Hindoos. 

The  methods  of  using  fractional  and  negative  exponents 
were  determined  by  Wallis  (1659)  and  Sir  Isaac  Newton. 

The  three  progressions  were  first  used  by  Pythagoras  (569 
B.  C.-500  B.  c.) 

Permutations  and  combinations  were  investigated  by  Pascal 
and  Fermat  (France,  1654). 

The  binomial  theorem  was  discovered  by  Newton  (1655), 
and,  as  one  of  the  most  notable  of  his  many  discoveries,  is 
said  to  have  been  engraved  on  his  monument  in  Westminster 
Abbey. 

Graphs  of  the  kind  treated  in  this  book  were  first  invented 
by  Descartes  (France,  1637). 

The  fundamental  laws  of  algebra  (the  Associative,  Com- 
mutative, and  Distributive  Laws;  see  Arts.  316-317)  were 
first  clearly  formulated  by  Peacock  and  Gregory  (England, 
1830-45),  though,  of  course,  the  existence  of  these  laws  had 
been  implicitly  assumed  from  the  beginnings  of  the  science. 

Students  who  desire  to  investigate  the  history  of  algebra  in  more 
detail  should  read  the  second  part  of  Fine's  Number  System  of 
Algebra,  Ball's  Short  History  of  Mathematics,  and  Cajori's  History 
of  Elementary  Mathematics. 


MATERIAL  FOR  EXAMPLES 


MATERIAL   FOR   EXAMPLES 

Formulas 

Formulas  used  in  the  following  subjects  may  be  made  the  ' 
basis  for  numerous  examples. 


I.  Arithmetic 

p  ==  br 
i  =  prt 
a  =  p-\-prt 

^  _  4(1  +  r)"  -  1] 
r 

II.  Geometry 

K  =  ibh 
K  =  iaVS 
K  =  ^h(b  +  60 
C  =  2itR 
K  =  7rR^ 
K  =  7rRL 
S  =  47rR^ 

T  =  TrRiR  +  L) 
T  =  2itR{R  +H) 

V  =  ■irmi 

V  =  i-TrR'H 

V  =  |,r7J3 

^~   180 

K  = 
V  = 

=  Vs(s  -a){s-  b)  (s  -  c) 
■■  \H{B  +  b  +  VBb) 

III.  Physics 

v  =  gt 

r 

s  =  vt  +  ^gf 

278 


MATERIAL  FOR  EXAMPLES  279 

^       2a  9-\-s 

^  4PP  '  l  =  i  +  i 

^      bh^m  f      P       V' 

Jl  H  =  MC^Rt 


=  ir\J 


g  E  =  ^^— 

^^R  C  =  f  (F  -  32) 

IV.  Engineering 

Tj  p   =  ^fcm    (horse-power  in  an  engine) 
33,000^ 

5  =  —  and  /'  =  /  +  If  (sag  in  a  suspended  wire) 
E  =  iri_  (elevation  of  outer  rail  on  a  curve) 
jy  =  BXD  J,  (^gight  a  beam  will  support) 

Ju 

^  ^  .0045(P-0  (T-t)C  ^i^^^^Yi  of  hot-water  pipe  to  heat 

a  house) 

T  =  ^         (tractive  force  of  a  locomotive) 


G  =  "^^^^^y^^  (no.  gal.  water  delivered  by  a  pipe) 

n  =  V     ^         (diameter  of  a   pump  to  raise  a  given 

amount  of  water) 

7)  ^  (/ K —     (diameter  of  balloon  to  raise  a  given 

^      ^  .5236(^-(?)  ^ 

weight) 


280 


SCHOOL  ALGEBRA 


Important  Numerical  Facts 

Areas 

Sq.Mi. 

Rhode  Island 1250 

New  Jersey 7815 

New  York   . 49,170 

Texas 265,780 

United  States .  3,025,600 

North  America 6,446,000 

Land  surface  of  earth 51,238,800 

Great  Britain  and  Ireland 121,371 

France 207,054 

Europe 3,555,000 

Astronomical  Facts 


Planet 

Diameter 

Distance  from  Sun 

Time  of  Revolu- 

Synodic 
Period 
in  Days 

in  Miles 

in  Million  Miles 

tion  about  Sun 

Mercury 

.   3030 

36 

88  da. 

116 

Venus 

7700 

67.2 

225  da. 

584 

Earth 

7918 

92.8 

365  da. 

Mars 

4230 

141.5 

687  da. 

780 

Jupiter 

86,500 

483.3 

11.86  yr. 

399 

Saturn 

73,000 

886 

29.5    yr. 

378 

Uranus 

31,900 

1781 

84  yr. 

369 

Neptune 

34,800 

2791 

165  yr. 

367 

Sun's  diameter 866,400  mi. 

Moon's  diameter 2162  mi. 

Moon's  distance 238,850  mi. 

Distance  of  nearest  fixed  star,  21   millions  of  millions   of 
miles  (or  3.6  Hght  years). 


I 


MATERIAL  FOR  EXAMPLES  281 

Dates  (a.  d.  unless  otherwise  stated) 


Rome  founded  .  .  753  b.  c. 
Battle  of  Marathon  490  b.  c. 
Fall  of  Jerusalem  .  .  70 
Fall  of  Rome  ....  476 
Battle  of  Hastings  .  .  1066 
Printing  with  movable 

type 1438 

Fall  of  Constantinople  1453 
Discovery  of  America  1492 
Jamestown  founded .    .   1607 


Declaration  of  Indepen- 
dence    1776 

Washington  inaugurated  1789 
Battle  of  Waterloo  .  .  1815 
Telegraph  invented  .  .  1844 
First  transatlantic  cable 

message 1858 

Telephone  invented  .  .  1876 
Battle  of  Manila  Bay  .  1898 


Distances 


From  New  York  to         Miles 

Boston 234 

Buffalo 440 

Chicago      912 

Denver .   1930 

San  Francisco   ....  3250 


From  New  York  to  Miles 

Philadelphia 90 

Washington 228 

New  Orleans 1372 

Havana 1410 

London 3375 


San  Francisco  to  Manila 4850 

New  York  to  San  Francisco  via  Panama 5240 

London  to  Bombay  via  Suez 6332 


Heights  of  Mountains 


Feet 

Feet 

Mt.  Washington  . 

.      ^6290 

Mt.  Mitchell  .    . 

.    .      6711 

Pike's  Peak   .    .    . 

.    14,147 

Mt.  Whitney  .    . 

.    .    14,501 

Mt.  McKinley  „    . 

.    20,464 

Mt.  Blanc  .    .    . 

.    .    15,744 

Mt.  Everest  .    .   . 

.    29,002 

Acongua  .... 

.   .    23,802 

282 


SCHOOL  ALGEBRA 


Heights  (or  lengths)  of  Structures 


Bunker  Hill  Monument , 
Washington  Monument 
Singer  Building  (N.  Y.) . 
Metropolitan  Life 

Building 700 

Eiffel  Tower 984 


221 

Olympic    .... 

.      882  ft. 

555 

Deepest  shaft  .    . 

.    5000  ft. 

612 

Deepest  boring    . 

.    6573  ft. 

Simplon  Tunnel  . 

.    12imi. 

700 

Panama  Canal     . 

.      49  mi. 

984 

Suez  Canal    .    .    . 

.    100  mi. 

Hudson 

Ohio 950  Rhine 

Colorado 1360  Amazon 

Missouri 3100  Nile    . 


Lengths  of  Rivers 

Miles  Miles 

280    Mississippi 3160 


850 
3300 
3400 


Rainfall  (mean  annual) 

Inches  Inches 

Phoenix  (Ariz.)  ...        7.9  New  York 44.8 

Denver 14  New  Orieans     ....      57.4 

Chicago      34  Cherrapongee  (Asia)   .    610 

Records  (year  1910) 

100-yard  dash 9f  sec. 

Quarter-mile  run 47  sec. 

Mile  run 4  m.  15f  sec. 

Mile  walk 6  m.  291  sec. 

Running  high  jump 6   t.  5f  in. 

Running  broad  jump .2    ft.  7J  in. 

Pole  vault 12  ft.  10|  in. 


MATERIAL  FOR  EXAMPLES  283 

100-yard  swim 55f  sec. 

1-mile  swim 23  m.  I65  sec. 

100-yard  skate 9|  sec. 

1-mile  skate 2  m.  36  sec. 

1  mile  on  bicycle 1  m.  5  sec. 

1  mile  in  automobile 27^  sec. 

1  mile  by  running  horse 1  m.  35f  sec. 

1  mile  by  trotting  horse  in  race     .  2 :03^  m. 

Throw  of  baseball 426  ft.  6  in. 

Drop  kick  of  football 189  ft.  11  in. 

Transatlantic  voyage  (from  N.  Y.)  4  da.  14  h.  38  m. 

Typewriting  from  printed  copy.    .  123  words  in  one  minute 

Typewriting  from  ney  material     .  6,136  words  in  one  hour 

Shorthand 187  words  in  one  minute 

Cost  1  lb.  radium $2,500,000 

Corn  crop  per  acre 255f  bu. 

Milk  from  1  cow  (1  year)   ....  27,432  lb. 

Butter  from  COW/ (1  year)  ....  1164.61b. 

Resources  (crops,  etc.,  year  1910) 

(All  these  figures  are  approximate  estimates.) 

Coal  lands  in  U.  S 400,000  sq.  mi. 

Coal  in  U.  S 2,500,000,000,000  tons 

Iron  ore  in  U.  S 15,000,000,000  tons 

Water-power  of  Niagara 7,000,000  H.  P. 

Natural  water-power  in  U.  S.    .    .  75,000,000  H.  P. 

Possible  water-power  in  U.  S.  (de- 
veloped by  stor:  ?e  dams,  etc.)    .  200,000,000  H.  P. 

Reclaimable  swai'  p  lands  in  U.  S.  .  80,000,000  acres 

Lands  in  U.  S.  rt Claimable  by  irri- 
gation      100,000,000  acres 


284  SCHOOL  ALGEBRA 

National  forest  reserves  of  U.  S.   .  168,000,000  acres 

Corn  crop  of  U.  S 3,000,000,000  bu. 

Wheat  crop  of  U.  S 700,000,000  bu. 

Cotton  crop  of  U.  S 13,000,000  bales 

Temperatures  (Fahrenheit) 

Normal  temperature  of  the  human  body 98.7° 

Ether  boils  at  96°  Temperature  of  arc  light  5400° 

Alcohol  boils  at  173°  (approx.) 

Water  boils  at  212°  Average  change  of  temperature 

Sulphur  fuses  at  238°  below  earth's  surface  1°  per 

Tin  fuses  at  442°  62  ft.  (increase) 

Lead  fuses  at  617°  above  earth's  surface  1°  per 

Iron  fuses  at  2800°  (approx.)  183  ft.  (decrease) 

Velocities 

Wind 18  mi.  per  hr.  (av.) 

Sensation  along  a  nerve  ....  120  ft  per  sec.  (av.) 

Sound  in  the  air 1090  ft.  per  sec.  (av.) 

Rifle  bullet 2500  ft.  per  sec.  (av.) 

Message  in  submarine  cable  .    .       2480  mi.  per  sec. 

Light 186,000  mi.  per  sec.  (approx.) 

Weights 

Boy  12  years  old 75  lb.  (av.) 

Man  30  years  old 150  lb.  (av.) 

Horse •  .    .    .    .  1000  lb.  (av.) 

Elephant 2^  tons  (av.) 

Whale 60  tons  (approx.) 

1  cu.  ft.  of  air 1|  oz.  (approx.) 

1  cu.  ft.  of  water 62.5  lb. 


MATERIAL  FOR  EXAMPLES  285 

Specific  Gravities 

Air ^0  0^    Stone  (average)     ...  2.5 

Cork 24    Aluminum 2.6 

Maple  wood 75    Glass 2.6-3.3 

Alcohol 79     Iron  (cast) 7.4 

Ice 92     Iron  (wrought)  ....  7.8 

Sea  water 1.03    Lead 11.3 

Water     .......    1         Gold 19.3 

Clay   ........    1.2      Platinum 21.5 


I 


Miscellaneous 


Heart  beats  per  minute  —  Frog 10 

Man 72      '^ 

Bird 120 

Smallest  length  visible  to  unaided  eye .    .    .  25T  inch 
Smallest  length  visible  by  aid  of  microscope    125,000  inch 
Accuracy  of  work  in  a  machine  shop    .    .    .  to  oT  inch 
Accuracy  in  most  refined  measurements  .    .  10.000.000  inch 
Dimensions  of  double  tennis  court    ....    78   X  36 
Dimensions  of  single  tennis  court     ....    78   X  27 

Dimensions  of  football  field 160   X  300 

Standard  width  of  railroad  track 4  8j 

Weights  and  Measures 

Avoirdupois  weight,  1  ton  =  2000  lb. ;   1  lb.  =  16  oz.  =  7000  gr. 
Troy  weight,  1  lb.  =  12  oz.  =  5760  gr.;  1  oz.  =  20  pwt.  = 

480  gr. 
Long  measure,  1  mi.  =  1760  yd.  =  5280  ft.  =  63,360  in. 
Square  measure,  1  A.  =  160  sq.  rd.  =  43,560  sq.  ft.;  1  sq.yd. 

=  9  sq.  ft,  =  9  X  144  sq.  in. 


286  SCHOOL  ALGEBRA 

Cubic  measure y  1  cu.  yd.  =  27  cu.  ft.  =  27  X  1728  cu.  in. 

Dry  measure,  1  bu.  =  4  pk.  =  32  qt.  =  64  pt. 

Liquid  measure,  1  gal.  =  4  qt.  =  8  pt.;  1  pt.  =  16  liquid  oz. 

Paper  measure,  1  ream  =  20  quires  =  480  sheets. 

Metric  system,  1  meter  =  39.37  in.;   1  kilometer  =  .621  mi. 
1   liter  =  1.057  liquid  qt.;    1   kilogram  = 
2.2046  lb. 
1  hectare  =  2.471  A. 

1  kilometer  =  10  hectometers  =  100  decame- 
ters =  1000  meters  =  10,000  decimeters  = 
100,000  centimeters  =  1,000,000  millimeters. 

At  the  option  of  the  teacher,  the  pupil  may  insert  on  the 
blank  pages  at  the  end  of  the  book  other  important  formulas 
or  numerical  facts,  particularly  those  which  are  important 
in  the  locality  in  which  the  pupil  lives. 


ANSWEES 

EXERCISE  1 

1.  56,  28.  4.  24,  12  quarts. 

2.  Daughter,  $8000;  son,  $4000.  6.  $12.40,  $6.20. 

3.  Man,  $96.60;  boy,  $32.20.  9.  11,250,000  bales. 

10.  Tenant,  $4000;  owner,  $2000. 

11.  New  York,  49,200;  Mass.,  8,200  sq.  mi. 

12.  200  and  40.  15.  .0036  and  .0009. 

13.  4.84  and  2.42.  16.  $90,  $30. 

14.  i  and  j\.  17.  4f  and  f . 

19.  Lowest  part,  12  ft.;  middle,  24  ft.;  top,  96  ft. 

20.  $1000,  $2000,  $3000. 

21.  Hat,  $7;  coat,  $14;  suit,  $21. 

22.  Niece,  $12,000;  daughter,  $24,000;  wife,  $48,000. 

23.  Cement,  3,375;  sand,  6,750;  gravel,  16,875  cu.  ft. 

25.  Nitrate  of  soda,  500;  ground  bone,  500;  potash,  1000  lb. 

26.  Lime,  190i^;  potash,  952^t;  sand,  2857/1  lb. 

27.  Boy,  $9.90;  adult,  $19.80.       30.  .0062,  .0124,  .0186;  ^^,  tV,  /i- 

28.  20,  40,  and  60.  31.  30,  30,  60,  120. 

29.  20,  40,  and  60.  32.  $94.74-,  $284.21+,  $1421.05+. 

33.  35^  lb. 

EXERCISE  2 

19.  (1)  6;  (2)  3;  (3)  h;  (4)  4;  (5)  8;  (6)  1;  (7)  9;  (8)  11;  (9)  10;  (10)  21. 
35.  Walter,  25  marbles;  brother,  35  marbles. 

37.  9  hr.  14  min.  42.  7  and  8. 

38.  15f,  12|.  43.  10,  11,  and  12. 

39.  35.  44.  121,  391  sq.  mi. 

40.  7,258.  "  45.  6290  ft. 


u 


SCHOOL  ALGEBRA 


EXERCISE  3 

16.  (1)  40;  (2)  72;  (3)  324;  (4)  9;  (5)  18;  (6)  26;  (7)  |;  (8)  9. 

17.  (9)  80;  (10)  125;  (11)  0;  (12)  4. 

23.  10;  32.  28.  2K 

26.  2,  4,  8,  32,  128.  29.  1200  sq.  ft. 

27.  81,  125.  30.  $24;  $14. 

31.  State,  $360;  county,  $720;  township,  $720. 

32.  Pedestal,  155  ft.;  statue,  151  ft. 

33.  Charcoal,  500  lb.;  sulphur,  500  lb.;  niter,  1000  lb. 

34.  800,000,000  bu. 


EXERCISE  4 

1.  17. 

16. 

16. 

31. 

I 

46. 

2+\^. 

2.  4. 

17. 

12. 

32. 
33. 
34. 
35. 
36. 
37. 
38. 
39. 

I 
1. 

f. 

I 
7. 

!f. 

1. 

24. 

47. 
48. 
49. 
50. 
51. 
55. 

0. 

3.  0. 

4.  2S. 

5.  7. 

6.  18. 

7.  2. 

8.  29. 

18. 
19. 
20. 
21. 
22. 
^. 
24. 

"57 
9. 
37. 
108. 

21. 
63. 

When  a 
When  a 
When  a 
When  a 

X 

:  =  3. 

;  =  2. 
J  =  2.  3. 
J  =  3. 

2x  +  l 

9.  6. 

1 

3 

10.  9. 

25. 

2. 

40. 

15. 

2 

5 

11.  3. 

26. 

15. 

41. 

i. 

3 

7 

12.  0. 

27. 

26. 

42. 

¥. 

5 

11 

13.  17. 

28. 

i 

43. 

i. 

i 

2 

14.  18. 

29. 

3. 

44. 

w. 

•       i 

li 

15.  12. 

30. 

3. 

45. 

4. 

1.5 

4 

EXERCISE  5 

1.  66;  60.32. 

8. 

402  ft. 

2.  180;  183.976. 

9. 

35°;  37f . 

3.  374.5; 

.3456;  105. 

10. 

1482f. 

4.  $37.50; 

$2052.05. 

11. 

78.54. 

5.  314.16. 

16 

Daughter, 

$14,400;  son,  $9,600. 

6.  10. 

17. 

$31.20. 

7.  257.28; 

100.5. 

18. 

Tenant,  $1890, 

owner. 

$2520. 

ANSWERS  ifl 

19.  Ownar,  $4125;  oUmt.  12475. 

20.  21,  105;  30,  90. 

21.  .0041,  .0231;  4)08,  .02. 

22.  Towoahip,  $10,800;  oouoty,  $5,400;  itoto,  $l,80a 

23.  GniYel,  2000  lb.;  «&nd,  1000  lb.;  cement,  500  lb. 


IXSRCI8I6 

2.ar. 

7.  r;  -U*. 

3.7*. 

8.  $9000. 

4.  5*;  -la^;  3(f. 

9.  -2f . 

5.  -4^;  7*;  -16!**. 

10.  $50;  -$25; 

-$50. 

15.  15  gAme0. 

15.  Defeated  candidate,  6,105;  wmniag  candidate,  6,315. 

W.  Walter,  30;  brother,  53. 

21.  f,  V. 

19.  81,  19. 

22.  234  mi. 

20.  IXn,  3.33. 

23.  555  ft. 

EXERCI8I7 

1.  3.                       6.  -  4. 

11.  13. 

16.  -5. 

2.  -2.                  7.  -2. 

12.  -9. 

17.  -2. 

3.  0.                       8.  6. 

13.  5. 

18.  62*. 

4.-2.                  9.-1. 

14.  -5. 

19.  +5. 

5.  -6.                 10.  -1. 

15.  -2.5. 

22.  Man,  240;  boy,  80. 

24.  Youafer,  $8.20; 

older,  $16.40. 

23.  $1880;  $1340. 

25.  $3.84  and  $8.84. 

26.  Ztne,  400;  tin,  800;  copper,  3400  lb. 
27.  3100  mi.  J».  22661  lb. 

29.  State,  $4000;  county,  $8000;  townihip,  $12000. 
30.  $3000,  $500a  $6000.  31.  612  ft. 


EXERCISE  9 

1.  -5. 

5.  5z. 

12.    -6a5«. 

18.  2:r. 

2.  -6. 

6.  -5o 

14.  4(aH-fe). 

20.  3o»  -  2a!». 

3.  2z. 

'    7.  72*, 

16.  Va+x. 

21.  a-^b. 

4.  -4j. 

11.  4eu:. 

17.  f^. 

22.  22^-llv». 

IV  SCHOOL  ALGEBRA 

^  23.  2hy^.  35.  2. 

24.  7x2  ^  22/2.  36.  x"  +  a:2  -  7a;  +  2. 

25.  w?  -mn-  2n«.  41.  $3000;  $6000;  $9000. 

29.  2x21/  +  4xt/2.  43.  11,  12,  13. 

30.  -  2x  -  4y  +  2z.  44.  25,  26,  27,  28. 

31.  -xy  +  2ax  +  y^-  3x«.  46.  716,555  sq.  mi. 

EXERCISE  10 

1.  4a6.  4.  8x.  7.  2(a  +  6).  11.  x'-  5x. 

2.  -4x.  5.  x2.  9.  V^Tx.  13.  3x3-7. 

14.  6x2  -}-  7a;  -  8.  22.  1  -  2x  -  2x3  +  x^  +  x«. 

15.  a  +  46  -  4c  +  d.  23.  m  -  3d  -  x  +  3c. 

16.  -  8  +  X  +  7x2.  24.   -  3x4  -f  3x3  +  43^2  _  e^.. 

19.  -l-2x  +  2x2  +  x3  + 3x4.  26.  -2x"»  +  4x»  -  x2 +  2. 

20.  12x?/2  -  x22/2  _  9a:2y.  .  27.  Gy. 

28.  3x;  -X  +  y;  -  3a2  +  2ab  -  ¥. 

31.  4x3  -  2x  -  2.  35.  3  in. 

32.  2x3  4.  6x2  _  2x  -  4.  36.  $10.10,  $14.70. 

33.  -  2x3  -  2x2  _  8x  -  2.        37.  $6.20,  $18.60. 

34.  4x3  -  4x2  +  8x  +  4.  38.  $1300,  $1600,  and  $2100. 

39.  $1200,  $1500,  and  $2300. 

40.  $3529tV,  $17641?,  and  $705H 

41.  $857f,  $17141,  and  $3428f. 


EXERCISE  11 

1.  5a  -  6. 

9.  4. 

17.  2c  -  6  -  cf. 

2.  X  +  1. 

10.  4x  -  1. 

18.  3x  -  2x3. 

3.  1  -  X. 

11.  0. 

19.   -7x3  +  x2- 

-2x-l, 

4.-1. 

12.  a  -  1. 

20.  2. 

5.  2x  +  1. 

13.  0. 

21.   -2y. 

6.   -x  +  Zy. 

14.  2  -  2x. 

22.   -3x. 

7.  1  -  2x. 

15.  X  +  1.     ,/ 

25.  i 

27.  3. 

8.  9x  -  1. 

16.  6. 

26.  5. 

28.  H. 

ANSWERS  V 

EXERCISE  12 

1.  3^-  (3x«  -  3x  +  1).  5.  x^  -  (-  4a;  +  rc«  +  4). 

2.  a  -  (b  -  c  -  d).  6.  a^b'  -  {2cd  +  c^  +  d^). 

3.  1  -  (  -  2a  +  a2  +  1).         7.  4x*  -  (9x2  _  i2xy  +  4y^). 

4.  1  -  (a2  +  2a6  +  ¥).  8.  x*  -  4x3  4.  43^2  _  (  _  4^  ^  4  _|.  ^.s) 
9.  (m  +  2)x  -  (n  +  4)2/  +  (3  +  n)^. 

10.  (1  -  a  -  6)x  -  (1  -  6  +  a)2/  -  (2  +  a  -  c)z. 

11.  (  -  7  -  2a  +  26)x  +  (12  -  c  -  Qd)y  -  (10  -  36)z. 

12.  (5  -  Scd)y  -  (3ac+  4a6  -  2c  +  5a)x  -  (5cd  +  4)«. 

13.  (3  -  a  -  c)x3  -  (2  -  a  +  c)x^  +  (1  -  2a  -  c)x  -  5. 

14.  (  -  a  +  1  -  26)x3  -  (1  -  6  +  a)x2  -  (1  +  a  +  36)x  +  3a. 

15.  (-  ¥  -  a3)x3  +  (a2  -  26^  -  c)x2  -  (a  -  3b  -  3c)x  -  a  -  c. 


EXERCISE  13 

1.  3. 

6.  2.74 

11.  -.3.                  17.  3i  SI 

2.  5. 

7.  1. 

12.   -  I.                   18.  4,  9. 

3.  3. 

8.  1. 

14.  f                       19.  3^  81 

4.  3. 

9.  h 

15.  12.                     20.  8.9,  7.5. 

5.  4. 

10.  fi 

16.  6.                       21.  8.9,  7.5. 

22.  8.9,  7.5. 

24.  13,  15,  17;  5,  7,  9,  11,  13. 

25.  18, 

20, 

22; 

8,  10,  12,  14,  16. 

EXERCISE  14 

1.  8. 

6.  3V3  -  3V2  -  1. 

2.  12. 

7.  5x3  +  x2  -  5x  +  2. 

3.  297.28. 

8.   -  2x»  +  3x^2/ +  7x2/«  -  22/». 

4.  36;  144. 

9.  12x  -  3. 

5.  2x*-3x'  +  2x2-7x- 

-2. 

10.  12x. 

11.  (1  +  a  -  2c)x5  -  (3  -h  c)x4  -  (1  +  a  -  3c)x»  -  (2a  +  5)x2  +  2. 

12.  1  +  2a  -  (1  +  2a  +  36  -  c)x  -  (1  -  2a  -  36)x2  -  (1+  2a  -  36)x». 

13.  6.  14.  3.  15.   -  i  16.  i- 

17.   -  8^2  4.  2oj:2  4- -Sac  +  2a«  -  a*. 


VI 


SCHOOL  ALGEBRA 


23.  Ihx^  -  llx. 

24.  fa:2  -  Ix  -  I 

25.  .95a2  +  .45a  +  .3. 

26.  1.23a2-h2.12a  +  .6. 

27.  Six +  y)  +  {y  +  z). 


18.  79. 

19.  x^-^Sax  +  5a\ 

20.  6x2  -  4ax  -  Qa\ 

21.  -4. 

22.  6,405,000  sq.  mi. 

28.  3a3  -  lOab  +  3a^¥. 

29.  First,  x^  -  x^  +  x  -  1;  second,  -  x"^  -{-  4:X  -  1;  third,  -  x^  -h  x 
+  14;  fourth,  -2:2  +  x-l. 

30.  First,  -  a^  +  2x^  -  3x  +  1;  second,  a:^  -  3x  -  9;  tMrd,  2x2-32; 
-6;  fourth,  2x^  -2x  -a. 

31.  2x3  _  2a;2z/  +  6x2/2  +  52/».  33.   -  2x3  +  2x2/2  +  3i/'. 

32.  4x3  _  2x^y  +  2x?/2  +  72/3.  34^  6^  _  2^22/  -  3y>. 

36.  17,480;  15,064  votes. 

37.  Lowlands,  44  mi.;  Culebra  Cut,  5  mi. 


EXERCISE  16 

1. 

-20. 

10. 

20a26cc?2. 

21.  2». 

40. 

0. 

2. 

6a. 

11. 

-  lS(^d\ 

23.  x^+\ 

41. 

0. 

3. 

-  15a6. 

12. 

16x32/3^4. 

24.  o2x»+«. 

42. 

16. 

4. 

-  30x22/2. 

13. 

.8x5. 

25.   -a3x«+». 

43. 

4. 

5. 

-8x2. 

14. 

ia3x«. 

27.   -a3x2»-«. 

44. 

1. 

6. 

15x2. 

15. 

.015x2. 

29.  10(a  +  6)s. 

45. 

3. 

7. 

-  12a2x2. 

17. 

Y^. 

32.  21(a  +  6)»'-H. 

46. 

iS. 

8. 

42x22/4. 

18. 

fx». 

33.  2"-i. 

47. 

2. 

9. 

-  21a2xy. 

19. 

2»»+i. 

35.  27.            39.  0. 

48. 

1. 

EXERCISE  16 

1. 

6a2x  +  9ax2. 

8.  2x"+3  -  3x"+-2. 

2. 

-  15x22/  +  10x2/2. 

9.   -  12x"+2  -  28x«+i. 

3. 

8x32/2  -  2x22/3. 

10.  3x"+i  +  3x". 

4. 

-  21a2fex2^  +  12a62a:2/2. 

11.  3x9«+5x7«. 

5. 

40a2c2n  -  15am^nl 

12.   -4a7'^  +  14a5^ 

6. 

—  7m3n  +  7m 

*n  +  21m5n. 

13.  x3  -  1.48x2  +  , 

.204 

X. 

7. 

24x32/2  -  15x22/3  _ 

3X2/*. 

14.  fx2  -  1x3  -  |x 

. 

ANSWERS  Vii 

15.  -  i^a»x3  +  i^aV  +  ia'x. 

16.  ^x*  +  lx'  +  iafi-  ix\ 

18.  10(a  +  by  -  6(a  +  b)^  -  10(a  +  b). 

19.  21  (x  -  yy  +  6(x  -  yy  -  18(x  -  yy. 

22.  56.  24.  5.567.  26.  56.  28.  80. 

23.  18.  25.  40.  27.  30.  29.  $238.25. 
30.  60.                             31.  169.9.                            32.  69.75. 

33.  Daughter,  $10,500;  son,  $5500. 

34.  Iron,  460  lb.;  aluminum,  158  lb. 

35.  19.  37.  -  9.  38.  3. 

EXERCISE  17 

1.  2x2  -  7x  -  4.  7.  32a5c  -  2a¥(^. 

2.  3x2  _  7a.  _  6.  8    33^^  +  x^i/^  _  143.^^ 

3.  2x2  -  9x  -  35.  9.  a'  +  63. 

4.  12x2  -  25x2/  +  12?/2.  10.  x*  -y. 

5.  28X4  +  X22/2  -  15?/4.  11.  8x4  -  2x3  +  x2  -  I 

6.  30x2i/2  -\-xy-  42.  12.  6x3  _  19^,22,  +  21x1/2  _  iq^^^ 

13.  2x6  -  5x4  -  2x3  +  9^2  _  7a;  +  3. 

14.  3x*y  -  10x3i/2  +  4x22/3  _|_  63.^4  _^  ^^ 

15.  x6  -  5x4?/  +  iQ^y2  _  iOa:22^  4.  5a:y4  _  yS^ 

16.  4x«  +  9x4  -  16^:3  _j_  22x2  -  21x  +  6. 

17.  x«  -  x*  -  7x4  +  3x3  +  173-2  -5x-  20. 

18.  x«  -  6x4?/  +  9x22/2  -  2/«. 

19.  a*  +  a'b^  +  64. 

20.  16x4  +  36x22/2  +  817/4, 

21.  X^  -  9x^2/2  +  7x42/3  _|_  13a^^  _  19^.2^5  4.  8^.^  _  y7, 

22.  -  x^  +  2ax4  +  8o2x3  _  160^3^.2  _  16^4^.  ^  3206. 

23.  a3  +  63  +  x3  +  3a62  +  3a26.  26.  Ax3  -  17fx  +  2. 

24.  0252  +  c2d2  _  a2c2  -  62(^2.  27.  .Ia2  -  .23a6  +  .126*. 

25.  ia2  -  162.  28.  4.5x3  -  7.1x2  -  .4x  +  .24. 

29.  x"+i  -  x«-i  -  6x"-2  -  2x  +  4. 

30.  X2"+1  -  X2"  -  2x2"- 1  +  3x2«-2  -  10x2"-3. 

31.  x"-4  +  x»-3  -  x"-2  +  x"-i  -  x"  +  7x«+i  +  lOx'H-*. 


Vlll  SCHOOL  ALGEBRA 

34.  2x<  +  5x3  -  8x2  _|.  iia;  _  20. 

35.  12x*  -  x3  -  27x2  -  3x  +  10. 

36.  6x«  +  5x*y  -  m:f?y^  +  Uxh/  -  6xy*  +  y^. 

38.  35.  40.  50.  42.  60.  44.  60. 

39.  35,  145.  41.  45,  50.  43.  24,  46.  45.  $200. 

46.  315,  85.  47.  $780,  $220. 

EXERCISE  18 

1.   -  a«.  5.   -  X  -  2.  8.  10x2  -h  7x  -  12. 

3.  (x  +  y)2  (x  -  yy.  6.  x*  -  x  -  2.  9.   -  5a  +  24. 

4.  16,  -  1.  7.  17x  -  12.  10.  24a  +  20. 

11.  x2.  21.  2x3  +  4x2  _  2x. 

12.  4x<-  12x3  ^  13^2  _  6x  +  1.       22.  x2  +  lOx  -  16. 

13.  40a  -  24a6.  23.  0. 

14.  -  20x.  24.  x2  -  5x  +  8. 

15.  -  6x2  +  I3x  -  4.  25.  3x2  _  lOxy  +  j/». 

16.  -  2x2  -  3x  +  6.  26.  x2  -  z^. 

17.  x«  -  7x  -f  6.  27.  4a2  -ax  +  bx  +  my  +  cy. 

18.  2a6  +  862.  28.  0. 

19.  2/2  -  4x2  +  2yz  +  z*.  29.  5a3  +  2a2x  -  llax2  +  10x». 

20.  2x  +  1.  30.  &xy. 

32.  -12.  35.-  12.  38.   -1.  41.  5. 

33.  0  36.-  18.  39.  -26.  42.  29. 

34.  12.  37.  16.  40.  1.  43.  8a«. 

44.  76p2.  52.  U,  i. 

45.  -  2a262  +  I4a6  -  5.  53.  $45,  $55. 

46.  27,  9.  54.  $10,  $45,  $45. 

47.  -  6a  +  296.  55.  $13^,  $28f ,  $28f ,  $28^. 

48.  50.  56.  80. 

49.  $100.  57.  22,  11,  17. 

50.  $920,  $80.  58.  13,  14,  15,  16,  17. 

51.  .0012,  .0003.  59.  15,  9. 

60.  Daughter,  $940;  each  son,  $1780. 
61.  25,  11.  62.  21,  22.  63.  $26,  $37,  $35. 


ANSWERS  ix 

EXERCISE  19 

1.  -  3.  4.  5xy.  12.   -  16a.  15.  .5a. 

2.  -  3^2.  5.   -  X.  13.   -  2iy.  17.  ^x. 

3.  -2a.  6.   -72/'.  14.  40m.  19.  2r2,47r,4r. 

21.  ^ ;  m;  2mt;.  22.   -  5(x  +  i/)';   -  10(:c  +  y). 

4  23.  .2(a  -  6)2;  .7(a  -  6). 

24.  a*";  a^";  -  a^^. 

25.  -  3a2;   -  6a;   -  6a';  2a4;  -  6a-»+». 
26.  a«-i-<;  a';  a"+».  29.  2«-2;  2"-^ 

33.  0.  34.  0.  37.  0. 

EXERCISE  20 

1.  -x^  +  Sx.  3.   -  26  +  3ac.  6.  1  +  m  -  m«  +  m». 

2.  5a;  -  2?/.  5.  3x2  _  2xy  -  y^.        8.  3x^  +  2x  -  5. 
9.   -  2a;2  +  Ax  -  30.  14.   -  2x  +  5a:2  -  3x2-». 

10.  .04a2  -  .08a6  -  1.662.  15.  x^  -  2:i^ -h  Sx^ -{■  x. 

11.  -fa;2+a;  +  ¥.  16.  -  2x^  ^- ^2?  +  x^  +  Zx. 

12.  -  fa'6  +  ia262  +  f a6'.  17.  3x"-i  -  2x"  +  4x"+i  _  a;n+2, 

13.  -  3a:2»  +  2a;"  -  4.  18.  -  5(a  +  6)  +  4. 

19.  (x  -  2/)2  -  .3(x  -  y). 

20.  X  - !/.  24.  60.  27.  300.  30.  $600. 

22.  x  +  l.  25.  120.  28.  $300.  31.  125  nickels. 

23.  90.  26.  84.  29.  600.  32.  $37.50. 
33.  14  quarters,  7  bills.                  34.  15  each.  35.  17  each. 
36.  2.                       37.  9.                     38.  2.  39.   -  2^. 

EXERCISE  21 

1.  3x  +  1.  4.  3x  +  7.  7.   -  5a;  +  8. 

2.  2a;  +  1.  5.  3a;  -  by.  8.  4a;  +  y. 

3.  4a;  -  by.  6.  3a  +  4c.  9.  a  +  26. 

10.  a;2  +  a;?/  4-  2/*.  14.  4a2x2  -  2axy'^  +  j/*. 

11.  9a;2  -  6a;  +  4.  15.  a;2  -  3a;  +  1. 

.  12.  3x  -  7.  16.  7a;2  +  8x  +  1. 

13.  25  +  20a;  +  16a;2.  17.  3a2  -  4aa;  +  xK 


SCHOOL  ALGEBRA 

18.  2y^  -4y^  +  y-  1.  21.  2:i^  -  x  +  1. 

19.  c*  +  cH^  +  X*.  22.  3x3  _^  4^2^,  _^  5^y2  _|_  2^3^ 

20.  2a;»  -  Sx^  +  4x  -  5.  23.  2x<  -  3x22/  -  2y\ 

24.  x3  +  2x^y  +  4x2/2  +  8^/3. 

25.  x^  -  2x3?/  +  4x22/2  -  8x7/3  _f.  ig?/*. 

26.  x^  —  x^y  +  ^y^  —  ^^y^  +  ^y^  —  y^' 

27.  64x6  _j_  iQ^yi  ^  4^.2^  4. 2^, 

28.  2x2  -5x-l.  39.  2x3  -  3x2  +  x  -  5, 

29.  3x2  -  X  -  5.  40.  la  -  ift. 

30.  2x3  -  4x2  -  X  +  3.  41.   la;  +  i^. 

31.  2x3  _|_  sx^y  -  4x1/2  _|_  j^.  42.  ia2  +  ^ab  +  ^62. 

32.  3a3  -  4a26  +  3a52  -  263.  43.  ^x2  +  ^x  -  I 

33.  x^  +  y^—  z^—  xz  +  xy—  yz.  44.  .4x  —  .5y. 

34.  c2  +  d2  +  n^-cd-cn-dn.  45.  2.4x  -  3. 

35.  y*-\-2y^  +  Sy^  +  2y  +  1.  46.  2x"  -  Sx'^K 

36.  2x*-  4x3  +  3x2-  2x  +  1.  47.  4x3"  +  3x2"  -  x«. 

37.  2x2/  -  2x2  -  3yz.  48.  4x"+i  -  3x"  +  x"-». 

38.  x2  -  3x  +  1.  49.  3x"-i  +  2x"-2  -  3x»-^. 

52.  X*  +  x32/  +  x22/2  +  x^  +  2/*  +    ^^ 


X* 


x-y 


53.  1  +  X  +  x2  +  x3  +  ^— -.  56.  X  +  5,  X  +  a,  X  +  2/. 

54.  1  +  ax  +  a2x2  + 58.  6  hr. 

55.  15,  3a,  3x,  3(x  +  2).  60.  4^  hr.,  36  mi. 

61.  108  mi.,  126  mi. 

63.  5f  hr.  after  second  boy  starts. 

64.  2  hr.  56  min.  after  second  train  starts. 


66.  6  hr. 

68.  4  mi.,  8  mi. 

67.  8  hr. 

69.  5  mi.,  10  mi. 

72.  A,  24 

mi.; 

B,  21  mi. 

EXERCISE  22 

1.  3.2x2  -  2.42x2/  -  -242/2. 

8.  6. 

2.  -  7.15a2  -  1.5a6  -  I.862. 

9.   -18. 

3.   -  3.8p2  -  .5p  +  3.85. 

10.  0. 

4.  2.6x2  -  .5x  +  2. 

11.  8x^-18x3- 13x2 +  9x 

5.  45. 

12.  1  +  2x  -  3x2  -  ^^ 

7.  -22X  +  54. 

13.  10,  11,  12. 

ANSWERS  XI 

14.  Cement,  400  lb.;  sand,  800  lb.;  gravel,  1600  lb. 


15.  9f  sec. 

25.  5a3  (x 

-yY- 

17.  a:«  +  X  -  1. 

27.   -5. 

18.  x2  -  3. 

29.  (1)  512;  (2)  , 

512;  (3)  64. 

19.  38. 

30.  6x3  _ 

22x2  _|.  loa;  +  10. 

20.  a  +  h]  3a-56  +  4c  +  7d 

32.  ix<-i^jax3+|a2x2  +  fa*. 

21.   -7-Sx  +  2y-. 

3,  etc. 

33.  x2"  +  ; 

rV  + 

y^\ 

22.   -2a +  36; 

-5. 

34.  2-3x  +  3x2- 

-.3x3  +  3x*.., 

35. 

X2- 

1. 

36. 

4.8x3 

-  17.95x2|/  +  18.45x2/2 

-  6.32/3 

37. 

x^- 

x3  +  2x2  ■ 

-  3x  +  5  +  .  , 

.  .  . 

38.  Qx-^y-  i 

42.   - 

3. 

39.  1.6x2  -  2xy  +  2AyK 

43.  2z 

—  X. 

40.  a2  +  62  +  c2- 

-ah- 

ac  —  he. 

44.  0. 

41.  2. 

48.  54 

■■;  2;  ^. 

EXERCISE  23 

1.  3. 

10. 

17  in. 

19.  3. 

28.  3. 

2.  4. 

11. 

-2. 

20.  1. 

41.  3. 

3.  10. 

12. 

11. 

21.  1. 

42.  2. 

4.  4. 

13. 

0. 

22.  14. 

43.  2|. 

5.  2. 

14. 

-I 

23.   -3. 

44.   -4. 

6.  3. 

15. 

10. 

24.  100. 

45.  2i 

7.   -5. 

16. 

6. 

25.  4. 

46.   -1. 

8.  4. 

17. 

-f. 

26.  1. 

47.  5. 

9.  10  ft. 

18. 

i 

27.  3. 

48.  7.8  ft. 

50. 

8.4. 

52. 

2yr.  3 

mo. 

EXERCISE  26 

1.  $63,  $21. 

5.  A,  $61 

;B,$39. 

2.  $48,  $36. 

6.  Owner 

•,  $45;  i 

3ther,  $22.50. 

3.  $48,  $24,  $12 

1. 

7.  23.22. 

4.  36,  12. 

8.  38. 

9.  Wife,  $9000;  each  daughter,  $3000. 

10.  Wife,  $12,200;  each  daughter,  $2200. 

11.  Alcohol,  50;  water,  62.5  lb. 


Xll  SCHOOL  ALGEBRA 

12.  20,  21,  22.  15.  91  sec.  18.  $26,  $37,  $35. 

13.  150.  16.  11,  24.  19.  73,  19. 

14.  21,  22.  17.  John,  36;  WilUam,  60.         20.  22,  11,  17. 

21.  Horse,  $67;  cow,  $27. 

23.  Limestone,  50;  coke,  350;  iron  ore,  400.  25.  12  lb. 

24.  87  yd.,  174  yd.,  99  yd.  26.  234  mi. 

27.  Passengers,  $1410;  freight,  $1675. 

28.  Niece,  $12,000;  daughter,  $16,000;  wife,  $36,000. 

29.  15.  33.  8  ft.  37.  T  X  12'. 

30.  4'  X  9'.  34.  5  yd.  38.  8  in.  X  12  in. 

31.  4  yd.  X  5  yd.  35.  20'  X  40'.  40.  36'  X  78'. 

32.  6  in.  36.  20'  X  60'.  41.  160'  X  300'. 

42.  Boy,  15  yr. ;  brother,  5  yr.  43.  Man,  20  yr. ;  brother,  10  yr. 

44.  8  lb.  51.  8  mm.  56.  6. 

45.  8J  lb.  52.  26'  X  34'.  67.  4,  5,  6,  7,  8. 

47.  13,  14,  15,  16,  17.        53.  30.  58.  8,  9. 

48.  19,  21,  23.  54.  |.  59.  8  yr.,  12  yr. 

49.  21  words.  55.  43.  60.  26|,  6i 

61.  27'  X  78';  36'  X  78'. 

62.  Mon.,  52;  Tues.,  104;  Wed.,  57;  Thurs.,  97. 

63.  $4.  65.  8  hr.  after  second  boy  starts. 

64.  3  hr.  after  second  boy  starts.     66.  38  da. 

EXERCISE  26 

1.  7*2  +  2ny  +  y'^.  4.  9x2  _  \2xy  +  4!/». 

2.  c2  -  2cx  +  x^.  8.  1  -  14?/3  _|_  49^^. 

3.  4x2  -  4x2/  +  y^-  9.  9x8  ^  30^7  -f  25x«. 

10.  36x*l/2  -  132x21/32^  +  1212/426. 

11.  25x2"  -  30x"y"2"'  +  92/2"22m^ 

12.  16xV°2*''  +  72x3?/3"+522n  _^  gl^^n^ 

13.  ix8  +  1x^2/  +  I2/2.  18.  2.25m2  -  .06m  +  .0004. 

15.  .04x2  +  .12x2/  +  .092/2.  19.  a^  +  2ah  +  fe2  +  ga  +  86+16. 

16.  .09a2  +  .024a62  +  .001664.  20.  a2  +  2a6  +  62  -  6a  -  66  +  9. 
23.  9  +  6a  +  66  +  a2  +  2a6  +  62. 

25.  4a4  -  4a26  +  8a2c  -j-  62  -  46c  +  4c2. 

26.  x2  +  2x2/  +  J'^-  2ax  -  2ay  -  26x  -  Ihy  +  a«  +  2a6  +  62. 


ANSWERS  XIU 

28.  998,001.  31.  2,601.  34.  992,016. 

29.  994,009.  32.  1,006,009.  35.  99,940,009. 

30.  99,960,004.  33.  9,409.  36.  9,840.64. 

EXERCISE  27 

1.  x^  -  zK  5.  X*  -4:.  9.  la^  -^b". 

2.  2/2  -  9.  6.  a^x*  -  ¥y\  12.  .0025a*  -  .096«. 

3.  9X2  _  y2^  8,   4a.2n  _  257/2"».  13.   ^zx^  -  .492/2. 

14.  a2«+2  -  162^-2  27.  a2  -  62  +  66x  -  9x2. 

15.  a?  +  2a6  +  62-9.  28.  x^  +  x'^y'^  +  y^. 

16.  x2  +  2xy  +  2/2  -  a2,  29.  a*  +  a2  +  1. 

18.  16  -  a:2  -  2x  -  1.  30.  4x*  -  29^2  +  25. 

19.  4x2  _  9^2  +  301/  -  25.  31.  4x4  _  2<dx'^y^  +  2/*. 

20.  a2  +  2a6  +  62  -  9.  32.  x*  -  3x2^/2  +  2/^. 

22.  16  -  x2  -  2x  -  1.  33.  a?  +  2a6  +  62  -  c2  +  2c  -  1. 

23.  4x2  -  92/2  +  302/  -  25.  34.  x*  +  2/*  -  xV  -  1- 

25.  x4  +  6x3+9x2  -4.  35.  x2  +  2x2/  +  2/2 -22 -22  -  1. 

40.  8096.  43.  999,975.  46.  9996  sq.  ft.  61.  996,004. 

41.  9991.  44.  1200.  47.  $8.96.  62.  999,996. 

42.  9975.  45.  292.40.  48.  $48.91.  63.  9409. 

64.  9991. 

EXERCISE  28 

1.  4x2  +  ^2  _^  1  _|.  4^.^  +  4x  +  2y. 

2.  x2  +  4^2  _|_  4^2  _  ^xy  +  4x2  —  82/2. 

3.  9x2  _|.  4y2  -I-  25  -  12x2/  -  30x  +  2O2/. 

4.  4a2  +  62  +  9c2  -  4a6  +  12ac  -  66c. 

5.  x2  +  42/2  +  922  _  ^xy  -  6x2  +  Vlyz. 

6.  16x2  +  92/2  +  1  +  24x2/  -  8x  -  62/. 

7.  x2  -  2x3  +  3x2  _  2x  +  1. 

8.  4a*  +  20a3  +  13a2  -  30a  +  9. 

9.  x«  +  2/2  +  2*  +  1  -  2x2/  +  2x2  -  2x  -  2yz  +  2y  -  2z. 

10.  4x2  4.  Qy2  4. 1622  +  25  +  12x2/  -  16x2  -  20x-  24yz  -30y  +  40z, 

11.  9x«  -  24x«  +  22x*  -  20x3  +  17x2  -  4x  +  4. 

12.  ix*  -  1x3  +  ^^a;2  -  ^x  +  25. 

13.  fx«  -  fx«  +  ^x*  +  ^x'  -  11x2  +  9x  +  36. 

14.  .04a2  +  .0962  4.  .25c2  +  .12a6  -  .2ac  -  .36c. 
29.  .0004x*  -  .012x3  _}_  .ii3;2  _  ,3^  _j_  25. 


Xiv  SCHOOL  ALGEBRA 

EXERCISE  29 

1.  x^  +  7x  +  10.  15.  a2  +  ia  -  i 

2.  x»  -  8x  +  15.  16.  a262  +  ^ahx  +  Sx^. 

3.  a:»  -  3x  -  28.  17.  a'b^  -  2ahx  -  3x^. 

4.  x^  +  4x  -  32.  18.  xV  -  4a;y22  _  2I3*. 

5.  x2  -  6a:  -  7.  19.  a:^"  -  25. 

6.  a:<  -  5^2  +  6.  20.  (x  +  ?/)2  +  8(x  +  y)  +  15. 

7.  x^  +  4x2  +  3.  21.  {x  +  ?/)2  +  2(x  +  y)  -  15. 

8.  a2  -  7ax  -  30x2.  22.  (x  +  2/)^  +  2(x  +  2/)  -  15. 

9.  x2  -  6x1/  -  7i/2.  23.  (a  +  26)2  _|_  8(a  +  26)  +  15 

10.  x2  +  .7x  +  .1.  24.  (2x+3?/)2-2a(2x+3!/)  -15( 

11.  x2  +  fx  +  h  25.  x2  -  a2  -  2a6  -  6*. 

12.  x2  +  5.02X  +  .1.  26.  4x2  -  a^  -  6a6  -  962. 

13.  a2  +  .52o  +  .01.  27.  4x2  -  a^  +  6a6  -  962. 

14.  x'  -  ix  -  i.  30.  381  ft. 

EXERCISE  30 

1.  2x2  _j_  11a;  _|_  12.  7.  5x2  _|_  34a.  _  7. 

2.  2x2  _  iix  +  12.  8.  3x2  +xy-  24y«. 

3.  2x2  -5x-  12.  9.  12a<  -  lla26  -  56«. 

4.  2x2  +  5x  -  12.  10.  |x2 .+  |x  +  i. 

5.  6a2  +  19a  +  15.  11.  2a2  +  .la6  -  .0662. 

6.  6a2  -  a  -  15.  12.  fx2  +  2ax  -  la\ 

EXERCISE   31 

19.  a«  +  2a6  +  62  -  x2  -  2xy  -  yK 

20.  a*  +  a2x2  +  x<.  36.  8a6.                         40.  3xy. 

22.  16a*  +  4a2  +  1.  37.  8x2-  gx  +  2.          41.   -  169x<. 


29.  a2  -  62. 

38.  - 

8x. 

42. 

c2  -  2c  +  2a. 

35.  2a2  +  862. 

39.  - 

6a +  5. 

43. 

x^y^-x^y-y^+y 

46.  72i. 

51. 

99001. 

56. 

38,025. 

61.  23.04. 

47.  3801. 

52. 

56.25. 

57. 

990,025. 

62.  9604. 

48.  39,800i. 

53. 

380.25. 

58. 

94.09. 

67.  -4^. 

49.  2401. 

54. 

9900.25 

59. 

96.04. 

68.  2. 

50.  2450i. 

55. 

5625. 

60. 

92.16. 

69.  9991  sq.  rd 

70.  9604  sq.  rd. 

71 

.  $35.96. 

72.  $80.75. 

ANSWERS  XV 


EXERCISE  32 


1.  a+x.  11.  .2x  -  ,Sy.  28.  54,  46. 

2.  3  +  2x.  15.  a;  +  1  -  a.  29.  53,  47. 

4.  5x  +  Qy\  16.  a  +  fe  -  2c.  30.  109,  91. 

5.  4x^  -  7y\  18.  a  -  6  -  c  +  1.  31.  10.9,  9.1. 

7.  a?>2  -  6c3d<.  19.  l-a-h+c.  32.  89,  71. 

8.  \x  +  fy.  27.  54,  46.  34.  12. 
10.  .5a  +  .46. 

EXERCISE  33 

1.  a2  -  2a  +  4.  10.  W  -iab^  +  iVfe*. 

2.  a;2  +  X  +  1.  14.  c2  -  c  +  ex  +  1  -  2x  +  xK 

3.  9x2  _|_  12a;  +  16.  15.  4  +  2x  +  2?/  +  x^  +  2x2/  +2/«. 

5.  25  +  5x3  _}_  a^.  16.  9x*  -  15xV  4.  252/6. 

6.  9a4  -  3aV  +  V^-  17.  a^  -  2a  +  1  +  ax2  -  x^  +  x«. 

8.  .04x2  +  2xy  +  2/2.  18.  x^  -  2x3  +  2x2  +  2x  +  1. 

9.  ix2  +  \xy  +  ly^.  19.  4x2  _  gxi/  +  4i/2  +  2x2  -  2yz  +  ^2. 
29.  23.                     30.  23.  31.  23.  32.  17. 

33.  53.  34.  47.  36.  90. 

EXERCISE  34 

1.  a*  —  a'x  +  a2x2  —  ax^  +  x*. 

2.  a<  +  a^x  +  a2x2  +  a7?  +  x^. 

3.  6«  -  l^y  +  W  -  632/3  +  62|/<  -  52/6  4. 2/J. 

4.  66  +  65y  _|_  5Y  +  632/3  +  52^  4.  5^/5  +  2/6. 

5.  a^  -  2a3  +  4a2  -  8a  +  16. 

6.  a^  +  2a6  +  4a4  +  8a3  +  16a2  +  32a  +  64. 

7.  X*  +  x3  +  x2  +  X  +  1. 

8.  x4-x3+x2  -x  +  1. 

9.  16x4  +  8X32/  +  4x22/2  +  2x2/3  +  2/4, 

10.  a}^  -  aH  +  a8x2  -  aV  +  aV-a^x^  +  a4x«-a3x'+a2x'  -ax«+x". 

11.  X8  +  X«2/3  +  a^^^  +  a;2^9  +  ^12. 

12.  81  +  27a  +  9a2  +  3a3  +  a*. 

23.  11.  ^  24.  13.  25.  12. 

26.  7.  28.  210. 


XVI  SCHOOL  ALGEBRA 

EXERCISE  36 

31.  x^+ax  +  a^+x  +  a.  42.  10. 

32.  x^ -\- ax  +  a^ -{- 5x  +  5a.  43.  6. 
35.  a;2  -  2ax  +  a^  +  5x  +  5a.  44.  1^. 

37.  (a2  -  62)2.  45.  _  4. 

38.  (a2  -  462)2.  46.  qq 

39.  (9x2  _  4^/2)2.  47  180,  540,  280. 

40.  3x2  +  14^  _  21.  48.  12i  mi. 

41.  -  11x2  4-  29x  -  13.  49.  617°. 

50.  120  ft.  per  sec.    Latter  is  liV  times  as  great. 

EXERCISE  36 

1.  x2(2x  +  5).  6.  3a2x2(a  -  5x).  9.  a^x{l  -  2x). 

2.  x(x2  -  2).  7.  9x4(2x  -  Sy).  10.  ix2(2x  +  1). 

3.  x(x  +  1).  8.  x2(l  -X-  x2).       11.  lah{3a  -  206). 

12.  ix3(5a  -  12x).  22.  (a  +  b)  (7x  +  5y). 

13.  .2x2(x  +  2a).  23.  xy{a  +  6)  [7x  +  5(a  +  h)yl 

14.  .02ax2(l  -  20a).  24.  7(a:  -  ym  -  2{x  -  y)]. 

15.  .6m(2n  -  m).  25.  3(2x  -  a)3I3  -  4(2x  -  a)2]. 

16.  3(a2  -  2ax  +  3x2).  26.  1694. 

17.  2x(l  +  2x  -  3x2).  27.  938.25. 

20.  2xV(2/'-  4x"2/  4-  3x2").  28.  58,190. 

21.  a"^63c2"(l  +  lie).  29.  314f. 

33.       }?      '  36.     ^  "^  ^ 


a  +  6  H-  c  3(x2  -  2) 

10       1  15  "^ 

31.  -^,  ^-  34. V"  .  ^  •  37. 


a  +  6    2  "20-6  +  30  *"' '  2(2x  -  3) 

^^•a  +  6  '^^•a  +  26  ^^-  2(2p2g2  _  i) 


EXERCISE  37 

1.  (2x  +  y)K  3.  (5x  -  1)2.  5.  c(7  +  26c)*. 

2.  (4a  -  32/)2.  4.  (x  -  lOy)K  6.  0^(6  +  2)\ 

7.  x(2/  +  1)2.  12.  2y{2a  -  5x)\ 

10.  x(2x  +  112/2)2.  13.  2x2(x  -  2)«. 

11.  a6(9a  +  76)*.  16.  (x"  +  j/)*. 


ANSWERS 

19.  (8a  ■ 

20.  (5x  ■ 

24.  (.2a  -  .36)2. 

25.  (5a  -  3x)2. 

26.  a  -  36. 

-12 
-5y 

-6)«. 
-  12xy)K 

27.  a  +  36. 

28.  1  -  2a. 

29.  2a  +  36. 

EXERCISE  38 

XVll 


17.  (a  -  6  -  c)«. 

18.  {3x  +  Sy  +  2zy. 

21.  (a  +  6  +  c)«. 

22.  {ix  +  Sy)\ 

23.  ihx  +  ly)K 


1.  (x  +  3)  (x  -  3).  9.  x(x  +  3a)  (x  -  3a). 

2.  (5  +  4a)  (5  -  4a).  10.  x^ix  +  3a)  (x  -  3a). 

3.  (2a  +  76)  (2a  -  76).        •  11.  m(l  +  8a)  (1  -  8a). 

4.  (x  +  2y)  (x  -  2y).  12.  2(11  +  x)  (11  -  x), 

7.  (1  +  ^m)  (1  -  8m).  15.  (a^  +  x^)  (a  +  x)  (a  -  x). 

8.  3(a:  +  2y)  (a;  -  2y).  16.  (a2  +  96^)  (a  +  36)  (a  -  36). 

17.  {7^  +  y')  (a:2  +  t/^)  (x  +  y){x-  y). 

18.  x(x2  -f  1)  (x  +  1)  {x  -  1). 

20.  x{a^  +  1)  (a2  +  1)  («  +  1)  (a  -  1). 

21.  (15x«  +  y)  (15a:"  -  y).  27.  (.9x  +  .056)  (.9a;  -  .056). 

22.  (fa:  +  ly)  (fa;  -  fi/).  28.  (a:^  ^  y^)  ^^  _  ^)^ 

23.  (fa  +  36)  (fa  -  36).  29.  (x^"  +  y^'z^)  (x^"  -  i/V). 

24.  (.3x  +  Ay)  (.3x  -  .4i/).  30.  {x +  y  ■\-\)  {x +  y  -  1). 

25.  (.la  +  .26)  (.la  -  .26).  31.  {x  +  y  +  l){x-y-  1). 

26.  (.52/  +  \h)  {.by  -  \h).  32.  (x  -  2/  +  3)  (x  -  2/  -  3). 

33.  (2x  -  21/  +  5)  (2x  -  2y  -  5). 

34.  (1  +  6x  +  I2y)  (1  -  6x  -  I2y). 

35.  (4x+22/  +  l)(22/-2x-l). 

36.  (11a  -  86)  (9a  -  26). 

37.  2/(xV  +  2^)  {x^y^  +  2*)  i^y"^  -  2*). 

38.  (9x«  +  4^2)  (3x3  +  2?/)  (3x3  _  2y). 

39.  x(x2  4-  Vly^)  (x2  -  121/23). 

40.  (a  -  6  +  2c  +  2)  (a  -  6  -  2c  -  2). 

41.  (10x2  -  lOx  -  9)  (-  10x2  +  lOx  +  11). 

42.  a  -  26.  43.  a  +  26.  44.  3  +  a.  45.  1+6. 

EXERCISE  39 

1.  (X  f  'i)  (x  +  2).  3.  (x  +  3)  (x  -  2). 

2.  (x  +  2)  (x  -  3).  4.  (x  +  11)  (x  -  4), 


XVIU  SCHOOL  ALGEBRA 

7.  (x  +  8y)  (x  -  2y).  20.  {x  -  8a)  {x  +  3a). 

8.  {x  -  8y)  {x  +  2y).  2L  (x^  -  S)  (x  +  1)  {x  -  1). 

11.  (x  -  9)  (x  +  4).  24.  a:(a;  +  4)  (x  +  3)  (x-  4)  {x-  8). 

12.  (x2  +  4)  (a:  +  3)  (x-3).  25.  (x"  -  8)  (a:»  +  7). 

15.  {x  -  12)  (a:  +  4).  26.  (a6  -  13c^)  (ah  +  2c2). 

16.  (x  +  16)  (x  -  3).  27.  (x  -i-a){x  +  6). 

17.  (x  -  24)  (a:  +  2).  28.  (a:  +  2a)  (x  -  36). 

18.  (x  -  12)  (x  +  8).  29.  {x  +  a-{-b){x  +  b  +  c). 

19.  (xy  -  12)  (xi/  -  11).  30.  {x  +  a-c)ix  +  b  +  c). 

31.  (x-y-6)ix-y  +  3). 


1.  (2a:  +  1)  (a:  +  1). 

2.  (3a:  -  2)  (a:  -  4). 

3.  (2x  +  1)  (a:  +  2). 

4.  (3a:  +  1)  (a:  +  3). 

5.  (3a:  -  5)  (2a:  +  1). 

6.  (x  +  3)  (2a:  -  1). 

7.  2x(a:  +  4)  (3a:  -  2). 

19.  (2a:  +  3)  (a:  +  1)  (2x  -  3)  (a:  -  1). 

20.  (3a:  +  2)  (a:  +  4)  (3a:  -  2)  (x  -  4). 

21.  (4a:  +  3^)  (3a:  -  4z). 

22.  2a:(6a:-t/2)(2a:  +  92/2). 

23.  (5a  +  46)  (5a  -  46)  (a^  +  6^). 

24.  (8a:2  -  9y^)  {2x^  +  y^). 

25.  iSx^  +  y){x^-Sy). 

26.  (5a  +  46)  (a  +  6)  (5a  -  46)  (a  -  6). 

29.  (a  +  fe  +  8)  (a  +  6  -  3). 

30.  (3a:  -  Sy  -  2z)  (x  -  y  +  Sz). 

31.  (3a:2  +  6a:  +  4)  (a:  +  3)  (x  -  1). 

32.  4a:(a:  +  4)  (a:  +  2)  (a:  +  1)  (a:  -  1). 

33.  2(1+ 3a:)  (2 -a:). 

EXERCISE  41 

1.  (m  -  n)  (m2  +  mn  +  n^).  4.  (a  +  26c)  (a2-2a6c  +  462c2). 

2.  (c  +  2d)  (c^-2cd  +  4(^2).  7.  (ah  +  1)  (a262  -  a6  +  1). 

3.  (3  -  a:)  (9  +  3x  +  a:^).  8.  (1  -  IQx)  (1  +  10a:  +  lOOx^). 


EXERCISE  40 

10. 

(2a:  +  5)  (a:  - 

•2). 

11. 

(4a:  +  1)  (3a:  ■ 

-2). 

12. 

(4a:- 

-  1)  (X  +  3). 

13. 

(5a:- 

-  1)  (:^  +  5). 

14. 

3a:(a: 

-  2)  (3a: 

+  1). 

15. 

2y{Sx 

^  +  2)  (a: 

-1). 

18. 

(8a- 

-  96)  (4a 

+  56) 

ANSWERS  XIX 

9.  xiSx  +  a)  (9x^  -  Sax  +  a^). 

10.  {8x  -  2/2)  (64x2  +  Sxy^  -\-y*). 

11.  a(l  +7a)(l  -7a  +  49a2). 

12.  (a  +  x)(a-  x)  (a^  -  ax  +  x^)  (a2  -{- ax  +  x"^). 

13.  (x2  +  y)  (x2  -  2/)  (x^  -  a:2y  +  y')  (x*  +  x2|/  +  y^). 

14.  (a  +  2n2)  (a  -  2^2)  (a2  +  2an^  +  4n*)  (a*  -  2an2  +  An'). 

15.  2x(5  -x2)(25  +  5x2  +  x4). 

16.  (2x2  _|.  y)  (4^4  _  2x'y  +  y'). 

17.  (a  +  &  +  1)  (a=^  +  2a6  +  62  -  a  -  6  +  1). 

18.  (5  +  26  -  a)  (25  -  106  +  5a  +  462  _  4^5  -f  a'). 

19.  (2  -  c  -  d)  (4  +  2c  +  2d  +  c2  +  2cd  +  d^). 

20.  (-2x  -  2/)  (13x2  -  5xy  +  2/2). 

2L  2x(2x2/2  -  32)  (4x2?/4  4.  Q^yi^  +  922). 

22.  (x  +  y)(x'-3^y  +  xY  -  xy' +  y'). 

23.  (x  -  y)  (x«  +  x^y  +  x^y^  +  s^y^  +  xY  +  xy^  +  t)^ 

24.  (a2  +  m2)  (a*  -  a2m2  +  m^). 

25.  (x4  +  ^)  (x8  -  x^?/^  +  7/«). 

26.  (a  -  26)  (a6  +  2a%  +  4a^62  +  8a363  +  16a26<  +  32a65  +  646«). 

27.  (a  +  x)  (a"  -  aH  +  a8x2  -  aV  +  a^x^  -  a^x^  +  a^x*  -  a^x^  +  a2x«- 
-  ax9  +  a:^"). 

31.  (3  -  x)  (81  +  27x  +  9x2  +  3^:3+  ^). 

32.  (4  -  o  +  6)  (16  +  4a  -  46  +  a2  -  2a6  +  62). 

33.  (2x  -  41/  +  1)  (4x2  _  16^^  +  iQyi  _  2a;  +  4i/  +  1). 

36.  (2x  -  a2)  (16x4  +  8a2x3  +  4a4x2  +  2a6x  +  a*). 

37.  {a^^-y'){a'-aY+'!f)' 

38.  (2X4  _|_  yi)  (4a;8  _  2x^'i^  +  ^). 

39.  [8x  -  (a  +  hy]  [64x2  +  8x(a  +  6)2  +  (a  +  6)4]. 


EXERCISE  42 

1.  (a  +  6)  (x  +  y). 

9.  {y'  +  1)  (2/  +  1). 

2.  (x  -  a)  (x  +  c). 

10.  (ax  -  1)  (x  -  2a). 

3.  (52/  -  3)  (x  -  2). 

11.  (X  -y){x-  3). 

4.  (m  -  2y)  (3a  -  4n). 

12.   (2  -  1)2  {z  +  1). 

5.  x(a  +  3)  (a  +  c). 

13.  (6  -  1)  (a  -  y). 

6.  (3a  -  5n)  (a  +  6). 

14.  (x  -  1)  (x2  +  2)  (x2  -  2). 

7.  x(x2  4-2).(x  +  l). 

15.  (x  ^y){a^  6). 

8.  2x(x  +  a)(x-'a)(x 

-1). 

16.  (xH-4)(x  +  2)«. 

XX  SCHOOL  ALGEBRA 

17.  {a  +  3)  (a»  -  3).  19.  {x  -  1)  {2x  -  1)«. 

18.  (x  -  y)  {2x  +  2y-  1).  20.  (x  -  1)  {x^  +  3x  +  3). 

21.  (x  -  2/)  (x  +  2/  +  x2  +  xy  +  2/'). 

22.  {x-y){x'-{-xy  +  y^  +  l), 

23.  (x  -  2/)  (2J^  +  a:i/  +  2/2  -  X  -  2/). 

24.  (x  -  y)  (x2  +  X2/  +  2/'  -  X  +  2/). 

25.  (x  +  2)  (1  +  a)  (x  -  2)  (1  -  a). 

26.  (x  -  y)  (x2  +  X2/  +  2/=^  +  X  +  2/  +  1). 

27.  4o(x-l)(x2+2). 

28.  (3a  -  x)  (3a  +  2x)  (a-  x).  32.  (2x  -  3)  (4x  -  3)  (x  -  2), 

29.  (x  -  2)  (x  +  3)  (x  -  1).  33.  (x  +  2)  (x  +  1)  (x  -  3). 

30.  (x  +  3)  (2x  -  5)  (2x  -  1).  34.  (x  +  3)  (x  -  2)  (x  -  4). 

31.  (2x  +  1)  (4x  -  3)  (x  +  1).  35.  (x  -  2)  (x  -  1)  (x  -  5). 

EXERCISE  43 

1.  {a  +  b  +  x){a  +  b-  x).  11.  {x  +  a-h)(x-a  +  b). 

2.  (a-  b  +  2x)  (a-  b-  2x).  12.  {x  -  a  +  y)  (x  -  a  -  y). 

3.  (a  +  X  +  2/)  (a  -  X  —  y).  13.  (a  +  2/  +  x)  (a  +  2/  -  x). 

4.  (3a  +  a;  +  2y)  {U-x-2y).  14.  (a*  +  x^  +  y)  (a^-  x»-  2/) 

5.  (4a  +  X  -  y)  (4a  -  x  +  2/)-  15.  {x  +  y  +  l){x-y-  1). 

6.  (m  +  X  +  2/)  (m  -  X  -  2/).  16.  {I  +  x  -  y)  {I  -  x  +  y). 

7.  (o  +  6  H-  2x)  (a  +  6  -  2x).  17.  {c -\- a -b)  {c  -  a -\-b). 

8.  (a  +  6  +  2x)  (a  +  6  -  2x).  18.  (a  -  6  +  c)  (a  -  6  -  c). 

9.  (a  +  6  +  2x)  (a  +  6  -  2x).  19.  (a6  +  1+  x)  (a6  +  1  ^  x). 
10.  (x  +  o  +  6)  (x  -  a  -  6).  20.  2{z  -  I  +  2«)  (z  -  1  -  2>). 

21.  (x  +  22/-5«)(x-22/  +  52). 

22.  (a  +  6  +  c  +  d)  (a  +  6  -  c-  d). 

23.  (x  -  22/  +  32  +  1)  (x  -  22/  -  3z  -  1). 

24.  (3a  -  26  +  5x  +  1)  (3a  -  26  -  5x  -  1). 

25.  (a  -  56  +  36x  -  1)  (a  -  56  -  36x  +  1). 

EXERCISE  44 

1.  {(?  +  cx-\-  x«)  (c2  -  ex  +  x«). 

2.  (x»  +  X  +  1)  (x*  -  X  +  1). 

3.  (2x»  +  3x  -  1)  (2x«  -  3x  -  1). 

4.  (2a'  -  3a6  -  362)  (2a2  +  3a6  -  36«). 


ANSWERS  XXI 

5.  (3a:«  +  Sxy  +  2y^)  (Sx^  -  2xy  +  2y^). 

6.  (7c«  +  9cd  +  5^2)  (7c2  -  9cd  +  Sd^). 

7.  (4x2  +  X  -  1)  (4x2  _  a;  _  1). 

8.  (10x2  +  X  -  3)  (10x2  -  X  -  3). 

9.  (15a262-f  Safe  +  2)  (15a2&2-  8a6  +  2), 

10.  2(4a2  +  6a6  +  62)  (4a2  _  6a6  +  62). 

11.  (a2  +  2a6  +  262)  (a2  -  2a6  +  262). 

12.  (1  +  4x  +  8x2)  (1  _  4^  _|_  8^.2) . 

13.  (x22/2  +  6x2/  +  18)  (x22/2  -  6x?/  +  18). 

EXERCISE  45 

10.  3x(x2  +  X  +  1)  (x2  -  X  +  1)  (x  +  1)  (a:  -  1). 

11.  (2a  +  1)  (a  +  1)  (2a  -  1)  (a  -  1). 

12.  2(x2  +  2x  +  2)  (x2  -  2x  +  2)  (x2  +  2)  (x2  -  2). 

13.  (x  +  9)  (x  -  5). 

14.  (2x  +  a  -  1)  (2x  -  a  +  1). 

15.  5a(x«  +  x3  +  1)  (x2  +  X  +  1)  (x  -  ]). 

16.  3x(3x  +  4)  (2x  -  3). 

17.  (X2  +XZ+  2^2)  (x2  -  X2  +  2^2). 

18.  (x  =fc  1)  (a  =fc  3).  25.  (a2  +  2)  (a  -  1). 

19.  (11  4-  x)  (10  -  x).  26.  2x(3x  +  1)  (a:  -  1). 

20.  (3x  -  52/)  (x  +  62/).  27.  (1  =*=  5z  +  z^). 

21.  7a(l  ±  a62):  28.  2(4  -  y)  (16  +  4^/  +  y^). 

22.  2(3x  +  4)  (x  +  1).  29.  (1  +  o  +  6)  (1  -  a  -  6). 

23.  (x  +  2)  (x-  1)  (x2-x  +  2).  30.  (3a  -  5)  (7a  +  6). 

24.  3a(l  +  a)  (1  -  a  +  a2).'  31.  (x^  -\.  y^)  {x^  -  3^  +  y^), 

32.  (2x  +  9z')  (4x2  -  18x23  +  8l2«). 

33.  45x4(3^/2  =t  1). 

34.  (a3  +  5)(a  +  2)(a-2). 

35.  (c  +  d  -  1)  (c2  +  2cd  +  (i2  +  c  +  d  +  1). 

36.  (x-2/)(x-2/  +  2). 

37.  (8x  -  92/)  (3x  +  42/). 

38.  (x  +  2y)  (x  -  2y)K 

39.  (a  +  3)  (a  +  2)  (a  -  3)  (a  -  2). 

40.  (z2  ±  z  +  1). 

41.  (a  +  6  +  ^)  (a  +  6  -  c)  (a  -  6  +  c)  (a  -  6  -  c). 


XXli  SCHOOL  ALGEBRA 

42.  {7x  +  ^y){^x-7y). 

43.  (2  +  n)  (16  -  8n  +  4n2  -  2^3  +  n^). 

44.  5x(x2  +  2/=)  (a:^  -  a:^  +  y^). 

45.  (m  +  n)  (m^—  m^/i  +  m^n^—  m?n^  +  m^/i*  —  mn^  +  n*). 

46.  ia(2a;  +  2/)  (4x2  -  2xy  +  y^).  49.  (2a  ±  1)  (a  =t  3). 

47.  (1  +  x2)  (1  +  xy  (1  -  a;).  50.  {x  ^  2)  (x^  ±  2a:  +  4). 

48.  6(x  -  3)  (4  -  x).  5L  {x  +  l){x^  2)  (x  -  3). 

52.  y{2x  -  02)  (16a:4  _|_  33^22  +  4^224  +  2xz^  +  28). 

53.  (x4  +  6a:2^2  +  2/4)  (^^  +  y^  (^^  _  2^)2. 

54.  (x2  +  2/2  +  22)  (re  +  2)  (x  -  2). 

55.  (aa:-2/)(a:-l)(a:2+x  +  l). 

56.  (a  +  &)(a-76). 

'  57.  (a-6+  a:-2/)  ia^-2ab  +  62-aa:  +  &a;  +  a2/-^2/  +  x^-2xy+  y^). 

58.  a262(o  -  6)2.  60.  {x  +  2/)  (2a:2  +  xy  +  2y^). 

59.  (a:<  +  a3)(a;8-a3a^_|_a6).  gi.  (^-6)  (a:  +  2/)  (a -6  +  a:  +  2/). 

62.  (a  -  6  +  2a:  +  22/)2. 

63.  (a2  +  1)  (a4  -  a2  +  1)  (a  =fc  1)  (a2  ±  a  +  1). 

64.  (2a  -  36)  (2a  +  36  +  2).  68.  (a  =t  52)  (1  -  a:)  (1  +  x  +  a:2). 

65.  (2a  +  36  +  1)  (2a  -  36  -  1).  69.  (3  +  a)  {x  ^  3). 

66.  {x  +  2/)2  (a:  -  y)\  70.  (a  +  6)^. 

67.  {x  -  1)2  (a:  +  2)2.  71.  (a  -  x)\ 

72.  (a6c  —  mnp)  {ax  —  my). 

73.  (a:  +  2  +  2a  -  n)  (a:  +  2  -  2a  +  n). 

74.  (a:  -  2)  (a:  +  5)  (2a:  +  1).  76.  ^x-'y^x  +  yY  {x  -  y)\ 

75.  (a2  +  262)  (a2  -262  +  1).  77.  2  (9a:  -  y)  {x  +  3y). 

78.  (1  +  a:  -  a:2)  (1  +  2a:  +  2a:2  +  ar»  +  x*). 

79.  (1  -  x)2  (1  -  2a:  -  a:2).  82.  (a  -  36  +  3)  (a  -  36  -  3). 

80.  (a:  +  1)  (ax  -  c).  83.  (a:2  +  4)  {x  +  2)^  (ar  -  2)2. 

81.  (a:2  =t  9a:  +  1).  84.  9(1  ^  x)  {7x*  +  6a:2  +  3), 

85.  (a:2  -  3  +  7y)  {x^  -  3  -  7y). 

86.  (a:22/2  -2+2x-y)  (x^y^  ^2-2x  +  y). 

87.  {ax  —  bmy)  {an  +  cmz). 


EXERCISE  46 

1.  2,  3. 

3.  3,  4.                    5.  4,  -  3. 

7.   =^=3. 

2.  2,-1. 

4.  3,  -  2.               6.   =t  4. 

8.  0,  ± 

ANSWERS 


XXIH 


9.  0,  =fc  5.  13.  0,  3,  -  2.      17.   -  1,  ±  2. 

10.  0,  =t  3.  14.  0,  -  2.  18.   -  1,  ±  3. 

11.  1,  h  15.  0,  -  a.  19.   =t  1,  =t  2. 

12.  2,  -  f .  16.  0,  =t  o.  20.  1,    =t  2. 
25.  h  1.                       26.  2,  2. 

28.  x2  -  7x  +  12  =  0. 

29.  x2  +  3a:  -  10  =  0.      . 

30.  x^  +  10a:  +  21  =  0. 

34.  -  5,  9.  36.  3,  -  4. 

35.  4,  -  10.  37.  0,  9. 

42.  50'  X  150'.  43.  617°. 


31.  x^ 

32.  x^ 


21.  -  1,  f. 

22.  ±  1. 

23.  0,  =t  3. 

24.  1,  2. 
27.   =«=  2,  ±  3. 
-  4a:  +  4  =  0. 


2a:  =  0. 


33.  r'  -5x^-hQx  =  0. 

38.  0,  25.  40.  8,  9. 

39.  0,  ±  5.  41.  I,  -  2. 

44.  49,170  sq.  mi. 


1.  2ab. 

2.  5x% 

3.  8a2x2. 

6.  a:  -  3. 

7.  a:(2a:  +  3). 

8.  a  —  x. 

9.  a:  -  1. 

10.  a(2a  +  1). 


EXERCISE  47 

11.  a:  +  l. 

12.  4aa:(a  —  x). 

13.  x{x  -  1). 

14.  2a:  -  y. 

15.  a:(a:  -  2). 

16.  6(1  -  a2). 

17.  1+a  +  aK 


20.  0262(a  _  5)8. 

21.  3a:2(a:  -  y^. 

22.  a:  -  ?/. 

23.  (a:  -  y)^. 

24.  2  -  X. 

25.  a  -  3. 

26.  a;(a:  -  a). 


EXERCISE  48 


1.  6a262. 

2.  SQa^xY- 

3.  12a6c. 

4.  12a262c2. 

7.  2a:(a:2  -  1). 

8.  6ah{a  +  6). 

9.  14a:2(a:  -  3). 

10.  (a:3  -  1)  (x  +  1). 

11.  (a:2  -  2/2)  (a:  -  2y). 

12.  6a:(a:  +  1)  (a:  -  1)2. 

13.  15  abx^yix  -\- y)  (x  -  y)^. 

14.  a:(a:  +  5)(a:-8)(a:--l). 

15.  a«  -  ¥. 

16.  6a:2(a:  +  1)  (a;  -  1). 


17.  12a6(a  +  6)  (a  -  b). 

18.  (2a:  +  l)(a:  +  l)(2x-.l). 

19.  6a:(a:3  _  i)  (-^  _  i). 

20.  6a:(3a:  +  10)  (2a:-7)  (a:-3). 

21.  2a:(l  +  x^)  (1  +  a:)  (1  -  x). 

22.  uxY(x  +  ly  {x  -  ly. 

23.  Q^^iSx  +  1)  (a;  -  1)  (3a:  -  1)*. 

24.  {x  -  1)2  (x  +  1)2  {x  +  3)2  (x-3). 

27.  a'¥(a  +  by  (a  -  by. 

28.  lSa^¥(^{c  ^d)(a-  dy. 

29.  a*bHa  +  6)2  (a  -  by. 

30.  36a:4(a,  4. 2^)2  (3.  _  y)3. 

31.  a-b. 

32.  36(a  -  by. 


Xxiv  SCHOOL  ALGEBRA 

33.  (a  +  b)(a-  b)  (x  -  y).  35.  {x  +  1)  (x  +  2)  {x  -  2)K 

34.  (a  +  6)  (a  -  6)  (x  -  2/)3.  36.  x'ia  +  x)  {a  -  x)\ 

37.  2,  3. 

39.  Irrigable  land,  100,000,000  A.;  swamp  land,  78,000,000  A„ 

40.  2  da.  14  hr.  40  min.  41.  866,400  mi. 

42.  Son,  $3600;  daughter,  $5600;  wife,  $10,800. 

43.  2162  mi.  44.  39.37  in. 


EXERCISE  49 

^    a  +  6  +c  .    6c 

A'    7i •  O. 


3  "•  o 

I 
43,560' 


3    -^—.  8  -^^-t^. 


4. 

2a 
Zx 

5. 

4x 
by' 

6. 

X 

2-3ax 

7. 

Sxz 

4^2 

8. 

3a 
462* 

9. 

1 

2a -1 

10. 

1 
2a* 

11. 

1, 

a 

12. 

x-y^ 

1 

EXERCISE   60 


1.  I  13.  3^ 


2a 
3a;* 

14   2(a;  +  l) 

15. 
16. 
17. 


3 
5 

2ix  -  y) 

a-hh 

2(a  -  b) ' 

2 


3x  —  4y 


-i 


19. 
20. 
21. 


.  22. 

x+y  x-y 


2x  +  3y 

7x-hSy 

2x2 

x2  +  3a;  4-  9 
a;-3 
1 


^u. 

2a;  +  3y 

24. 

2x  +  y 
2x-y 

25. 

a  +  b  —  e 

a  —  b  —  c 

26. 

1+a-x 

a;  +  a-l 

27. 

3a;  +  4a 
x-\-  a 

28. 

x-2 

y* 

29. 

a;  +  3 
a;  +  2 

30. 

x^-y^. 

31. 

x-y^ 
a  +  b 

oo 

x-y-z-2 

ANSWERS  XXV 


EXERCISE  61 

2.^.  ^.-Z^.  10.-5^. 

3.  y-z^.  7.  - 1.  11.  ^  +  %+'- 

y  +  2x  X  -3 

A    3+m                                 -1  IP    2  +  a  +  b 

4— m                           o  +  0  4- c  2  — a  +  o 

18.  f           19.  6.  20.  |.          21.  4.          22.  20.            23.  5. 


EXERCISE   52 

1.  6f.  2.  13f.  3.  lOJrf 

4.x-2  +  ^-  14.x-l-^2. 

5.  2x^  +  3 -A.  15.  3a+      ^"^ 


2a;  '  3a2  -  2b 

6.  2a2x2  +  l -y^-  16.  x2-2a;  +  2+      ^ 


5ax  '  X  +  3 

7.  x2  -  4x  +  5 T-r-  17.  2a  -  26  + 


x+1  ' a+6 

n.^_,  +  2-4^^^.  20.1 -.  +  2x^-3^-2-' 


X^+X-1  '  '  1+X-X2 

12.  x»+:»'H-2x  +  3+  !""  "^  V      21.  4-2x  +  3x2       So;' -  3a:* 


X2-X-1  '  2  +  x-a:2 


EXERCISE  63 

1.  ^.  ^    a:^-2re  _        a:»  -  g 

2.  i^.  X  —  1  x^  +x  +  1 

,    a'-a  +  l  7    8x2 -y,  o' +  1 

4-  ^^ ^'   2x  +  l  ^^'  a-2' 

7?  Q   q2  +  Q6  .     x»4-xy 
X  —  1                               a  +  2o  X  -t  a 


XXyi  SCHOOL  ALGEBRA 

,„    2hc-¥+c'-a*  ..    (a  -  b)^  .a    ^  -  I 

i-o.    ^rr •  10.    -: •  lo.    ; — — 

26c  4  X  + 1 

14.(2  +  30)!.  17.^.  1,       ^ 


4  l+x                  -'  x-1 

20.  3248  mi.                                  2L  27^  sec. 

EXERCISE  64 

1.  if,  if.  -    _a 46_    2ab_ 

2.  tiH,n.  •  2026=^' 20=^62' 2a262- 

o    4a;  15x  ,,1        2a^  -  2a      3a 

"•    To»  "To"  • 


18'  18  a2  -  a     a^  -  a  '  o^  -  a 

jQ       x»  +  x2  +  a;  X  +  1 


(x  +  1)  (x'  -  D'  (X  +  1)  (x»  -  1) 
jj  X  x(2x-3)      4x2-9 


15. 


x(4x2  -  9)'  x(4x2  -  9)'  x(4x2  -  9) 
2x  +  4     15x  -  30         18 
6(x2-4)'6(x2-4)'6(x2-4)* 


EXERCISE  56 


1    1?.  7    3x  +  l  j^  3w2+l 


6x  24  ^"  (m+  1)  (m  - 1)2 

8x  -  9  +  12a  8.  1.  ^^  {a  -  36)« 

12ax    *  9   4x  •  4(a  -  6)2* 

_  15b  -  4c  -  6a  '  1  -  f  16.   ^'  . 

3-  6^6^ 10.^^^.  ^^-4 

,  ,  ,  ,      .  3x2  2x2  +  3x  -  1 

4.  ?^.  jj  25a -20b  17-   x(x2-l)  ' 

^^^  ^^-    12  18.  0. 

g  9  +  10ax2  y8-3x2g»-6y02  1 


12ax2 


^^'  6x22/22         •     ^^-  8x2-2 


6.^^.  13.-J?£-.  20.^. 

o2  —  o2  x*  —  4  x2  —  1 

01  x2  +  5x  + 10  04  5^'y  -  ^y" 

^"^     (x  +  1)  (x  +  2)  (x  +  3) '  ^^-  x(x2  -  t) '  25.  0. 

5x(x  +  3)  x2  +  4x  -  13 

^  •  (2x  +  l)(2x-l)(x  +  l)*  2(x2-l) 

23         ^'  27    ^-^^ 

(a  +  6)»  X3  +  64 


ANSWERS  XXVn 


28.  ^^  +  ?Q^-l..  29.  -1 


6(x2  -9){x-  3)  (x  -  3)  (x  -  4) 

Qn   ^'^  +  2x  -  1  ^^  o-3b  36.  0. 

^^-       x^-1  ^^'  ¥^^^'  13 

31.  0.  3xy  *"'•  8(1  -a«) 

K  42/2  _a.2- 

32.  r-^-  „.  ^  38. 


X 


1  -  x2  35.  0.  1  -  x2 

39.  .        ...^-1. ^.  43.    ^^  -  ^^^  -  1^ 


40. 


{x  -  2)  (x  -  3)  (x  -  5)  (x2  -  9)  (x  -  1) 

5-46 44.  0. 

(a  -  3)  (a  -  2)  (6  -  2)*  ITx^  -  42x  +  39 

41.  0.  15(x2  -  9) 

42..^   7^,   ..  46.  ^^-4^-22 


12x(x  +  1)  (x  -  2)  (x  -  3)  (x  -  5) 

47.  0.  49.  0.  ^^  1 


48.  1.  50.  1.  51.  0.  ^i^  + 1) 


EXERCISE  66 

263x  a  +  x  20.  1. 


Sacy  x^{a  —  x) 


21.  1. 


2.  ^^.               ,         lo   g'  +  g  +  l  .. 

4x2  3.  1.         12. 22.   -— . 


a 


a 


4-   --V"  IM    fa+1)  (2x-l)       „„        , 

^-  ;t(2x  -  1)'  ,,        2  24.  "  +  ''"'?• 

„6  ^*-^TT-  «--  +  ! 

2^^'  15  X  25.  1. 

»«  +  2x  -  3  '3^-xy  +  y^'  g'c  +  g^  +  6c» 

X  jg^  a4-64-c 

8   -^LzJ_.  1  27.  1. 

''•  a(x  +  1)  17.  -\. 

2a; +  3  28.  i- 

^-  3(3:c-l)'  18-  1- 

10.  ^£  +  l|.  19.  -^.  29.  2^i^. 

(x  + 1)2  x  +  y  ^mn 


^^XVlll  SCHOOL  ALGEBRA 


EXERCISE  57 


-    2(2  -  ar)                                  i  o      ^2   .  m 

I.  -^ -'                   12.  — i— .  OQ    2  -  q2  +  6^ 

^                                 2x2  -  1  ^'^'  2 

^^  +  1                               «  +  1  2a  +  26 

^*'^"^^*                         14    e^_±^^_Z_^.  25.   -1. 
•  .    6                                    *  ac  4-  6c  +  a6 

.,    x-a  +  1  26.^^±1±^, 

5  o-J^                        1^-  ^T^^-  26c 
'    *  a  +  1                          16.  0.  27.  i. 

6  ^^-2.                      17    a6-cd  +  l  '  ""' 

18                                 a6-cd-l*  28.  2x. 

8.  -  4±1-.              19    <^  +  1)  30.   -  1. 

9.  1.                              20.  ?^^.  ^^-  rT2i- 
^                                          a 

10.  a+a:.                        21.  a  -  1.  ^2.  14. 

11    L±y.                       22   -^  ^^-   ~  ^^• 

""-y                             '^-x  34.102. 

EXERCISE  69 

1.  3a  +  56  -  4c. 

2.  -5-a-2x  +  y-^z;0-a~2x  +  y  +  z. 

3.  9a2  +  62  +  c2  +  4c^  -  6a6  -  6ac  +  12a(i  +26c  -  46./  -  4cd. 
7.   -8a2  +  20a+9.  19.   -  ^. 


8-  35.  o^  a  +  1 

9. 


23a  2^-  "3^ 


^  21.  5  +  2a  -  3a2. 

^^-  2'  ^'  4x3  _  26x 

12.  32,  12,  lOf.  28.  ^^,_^^^^,_^y 

^^'  ^^'^^'''-  OQ  4x3- 26x2- 26x  +  144 

14  (J^SI^,  ^^'  '     (x2-4)(x2-9)   ' 

18.  f,  -A.  30.-1^,. 


ANSWERS 

X 

31    ^(2x- 
^^-  2(3:.  - 

3) 
1)' 

38. 

1+rr* 

32.  0. 

39. 
40. 

1 

1. 

31                ^ 

**•  s(2  -  x)  (x 

-3) 

41. 

2x 
3' 

3«-(l-.)(.. 

4 

-  2)  (X  -  3) 

. 

42. 
43. 

5a:. 
_1 

X 

37.  L 

48. 

5240] 

oai. 

EXERCISE  60 

1.  2. 

17.  3. 

32.  ^. 

47. 

6f. 

2.  3. 

18.  10. 

33.   -5. 

48. 

12. 

3.  2. 

19.  ff. 

34.   -7. 

49. 

.5. 

4.  2. 

20.  A. 

35.   -3. 

50. 

.2. 

5.  -1. 

21.  5. 

36.   -4. 

51. 

15,200. 

6.  13f. 

22.  4. 

37.  2. 

52. 

.05. 

7.  5. 

23.  13. 

38.  Y. 

53. 

.05. 

8.  1. 

24.  ih 

39.  3. 

54. 

3  yr.  6  mo. 

9.  ¥. 

25.   -7. 

40.   -h 

55. 

122,  212. 

10.   -2. 

26.  4. 

41.  3. 

56. 

20. 

11.  5. 

27.  2. 

42.  8. 

57. 

14^. 

12.   -f. 

28.  I. 

43.  1. 

58. 

19i^. 

13.   -2. 

29.   -i 

44.  73. 

59. 

36f. 

14.  f. 

30.  -H. 

45.  5. 

60. 

10. 

15.  0. 

31.  0. 

46.  30. 

61. 

12. 

EXERCISE  61 

1.  -i 

5.   -9. 

9.   -f. 

13. 

5. 

2.  -3. 

6.   -I 

10.   -23. 

14. 

7. 

3.  -H. 

7.   -I 

11.  8. 

15. 

0. 

4.  12. 

8.   -2. 

12.  1. 

16. 

-3. 

17 

'.  i 

20.  183  ft. 

XXX  SCHOOL  ALGEBRA 


EXERCISE   62 

8.  2,  -  1*.  9.  5,  -  1*.  10.  2,  -2* 

11.  1*,  -J^.  12.  3,-1*. 

EXERCISE  63 

1-  3a.  a  17.  17a.  ] 

h  h' 

2.  -•      *  ^  18.  19a\ 

_«  n.S—t^  20.       ^ 


a-6^ 

0-& 

2c 

3-6 

5 -2a 

36  + 2d 

2a -c 

a6 

a  +  6  '  2a  +  b 

12.  ^.  21       ^^(^-^)    . 
2  a'  -  2a62-  fta 

13.  ^.  22  a^ 


14.  0. 


23.  0. 


7-  -^^— •  15.  3  24.   "^  +  « 


3a2-l 
0-6'   :  ■'''•  ab  +  bc  +  ac'  26.  ^1  -  2a  -  o»), 


8.  -^.  16.  ^^^ 


EXERCISE  66 

1-  4-  4.  2q  +  6  6.  5.  10.  24. 

2-12  ^  7.  59.  11.  30. 

,*           *  5^.  8-60.  12.  ^;f;A. 

3-   -  2.  >  9.  247.  13.  5V.        14.  45. 

EXERCISE  66 

1.  120.  3.  336.  4.  120. 

6.  Tin,  37A;  zinc,  75A;  copper,  301i^  lb.  6.  27,  28. 

7    1^   A    1156 
•  lOO'  20'   100  * 

8.  Owner,  $2800;  other,  $2000. 

9.  State,  $4000;  county,  $8000;  township,  $6000. 
10,  $350,  $420;  $330,  $440.  11.  33,  42. 


ANSWERS  XXXI 

12.  15,000,000,000  tons. 

13.  India,  234,375,000;  China,  421,875,000. 

14.  First,  $30,000;  second,  $22,500. 

15.  177^  cu.  ft.  22.  11.  29.  100  lb.  34.  U  da. 

16.  96.  23.  30  gal.  30.  80  lb.  35.  6  da. 

17.  860  million.  24.  li^  gal.  31.  |  gal.  36.  4  da. 
18.10.  26.  4flb.  32   1,  i.  37.  28|  min. 
19.8.  27.  9itlb.  '  9'  x'  38.  36  min. 
21.  60.  28.  150  lb.  33.  5f  da.  39.  169iV  min. 

4j     I  54t^  min.  past  4.  42     I  ^^^^  ^^'  P^^*  ^' 

1 38 A  min.  past  1.  '    1  54tt  min.  past  10. 

40     I  ^iT  a^d  38t\  min.  past  4. 
1 21 A  and  54iV  min.  past  7. 

44.  779  -  da.       46.  686  da.  48.  24  hr.  50.  6  hr. 

45.  398  +  da.       47.  15  hr.  49.  108  min.        51.  19ii  mi. 

f  1st,  each  12  hr.  63.  36  lb. 

g2     I  A,  42  mi.;  B,  40  mi.  64.  $20,000. 

2d,  each  492  hr.  g^    22^  min. 

^A,  1722  mi.;  B,  1640  mi.  ^,    ,  ,        ,  „„  «      . 

gg   g^^'  '     '  66.  5A  and  38A  mm.  past  10. 

54.  1100  ft.  ^^-  ^• 

55.  2357+ ft.  68.  29|yr. 

56.  gold,  21  lb.;  snver,  18f  lb.  69.  12  da. 
67.  Aluminum,  35  lb.;  iron,  45  lb.  70.  U  gal. 
58.  Copper,  51^  lb.;  tin,  48|  lb.  71.  6. 

^g     r  46f  bu.  oats.  72.  15. 

*'  ^  ^^3  bu.  corn.  73    ^p  +  2,  2p  +  3. 
f  $3250  at  4%.  ^5,       . 

^"-    I  $1800  at  5%.  74.  ^-q—  mi. 

61.  10;  14;  6;  24.  _abc_  .. 

62.  26,  27,  28.  ^^'  b-a 

EXERCISE  67 


15.  5i  yr.  17.  (1)  122°;  (2)  32°;  (3)  4892*^. 

18.  4 
20.  60  lb. 


16.  4^%.  18.  442  +  °;  617°;  1995  +  °;  2804*. 


Xxxii  SCHOOL  ALGEBRA 


EXERCISE   68 


1.  1,  1.  5.  2,  -  L  9.  I  |.  13.  15,  10. 

2.  1,  -  1.  6.  1,  -  h  10.   -  I  2.  14.  3,  -  4. 

3.  h  -h  7.   -  3,  U.  11.  3,  -  7.  15.  10,  -  10. 

4.  2,  3.  8.  i  -  I  12.  8,  9.  16.  12,  18. 

17.  7,  5.  18.  12,  21.  21.  Sugar,  6^;  rice,  5^ 

EXERCISE   69 

2.  1,  2.  4.  3,  -  2.  6.  3,  -  J.  8.   -  4,  3. 

3.  -  1,  -  1.         5.  2,-1.    .         7.  5,  4.  9.  2,  6. 

EXERCISE  70 

2.  3,  -  2.  4.  3,  -  2.  6.  1,  3.  8.  5,  1. 

3.  -  3f,  -  ^.  5.  3,  6.  7.   -  2, 1.  9.  ff,  4. 

10.   -  61,  -  3i  11.  ii,  ¥. 

EXERCISE  71 

1.  5,  12.  5.  3,  1.  9.  2,  4.  13.  18,  12. 

2.  5,  2.  6.  -  1,  4.  10.   -.2,  .6.  14.  9,  -  1. 

3.  i,  i.  7.  -  h,  h  11-  .015,  .01.  15.  17,  6. 

4.  -5,  -  1.  8.  1,  -  1.  12.  2,  -  3.  16.  2,  -  1. 

17.   -2,  -3. 

EXERCISE    72 

1.  2a,  —  a.  o    a     6 

8.    r>    -• 

2.  -6,  2a.  «>     " 

¥  -h         a -a'  9    1,  1. 

'^'   ab'  -a'b'  ah'  -a'h  '    <^    ^ 

^.  m-\-n,m-n.  10.  n-m,n+m. 

^    2b +  r    a-2  11.3,  ?^. 

6             a  0 

6.  a +  26,  2a- 6.  12.  a,  -  6.        13.  a  +  6,  a  -  6. 

'       cn  —  bd      ad  —  cm  -.     a  —  d    a  —  h 

'  on  —  bm     an  —  bm  '   b  —  d    b  —  d 


ANSWERS  XXxiil 

17.^ 
1  1  2 


15.  a,b.  --    a  +  2b     a  -  2b 

1  1 

16. 


a  +  6  +  c    a  +  6  +  c  18.   -  a,  b.  19.  b,  a. 

20.  a  +  1,  6  -  1. 

EXERCISE  73 

1.  1,  2,  3.  4.  2,  2,  2.  7.   -  3,  3^,  -  2„ 

2.  2,  3,  -  4.  5.  U,  1|,  U.  8.  2,  3,  1,  4. 

3.  3,  4,  7.  6.  2,  3i  -  4.  9.  12,  18,  -  24 

10.  a  +  6,  a  -  6,  2a.  11.  6,  40,  20. 

12.  X  =  -a+b  +  c.  , ,  3a  -  26 

,    ,  14.  X  = ^ 

y  =  a  —  b  -{-  c.  6 

2=a  +  6-c.  2/=  2a  +  36 

13.  X  =  a  -  6  +  1.  6       ' 
?/=— a  +  6  +  1.                                      _aH-6 

2  =  a  +  6  -  1.  "^  6 

.,_  6  +  c  +  c?  —  a  a  +  6  +  c?  —  c 

15.  a;  =—  — 


y 


4  4 

o  +  c  +  c?  —  6  a  +  6  +  c  —  d 


EXERCISE   74 


1.  -  1,  1. 

2.  i  -  i 

3.  i  iV. 

4.  i  -  i. 

10. 

1,1. 

11. 

12. 
13. 
14. 

h  - 1 

1,  -  h  h 

2,  -  h  1. 
1,  i  I. 

5. 
6. 

7. 

2n 

2n      . 

8.   i,  1. 

1+^2' 

a,  -a. 

1  -w2 

9.  ^,  ^. 
6     a 

ire     -2&C 

-  2ac.     -  2a6 
a-{-  c      a  +  6 

lb.   T~, ' 

26           2c 

I  -\-m  .  I  -]-n    m  -{-n 
1,1. 


EXERCISE  75 

1.  9,  14.  2.  9,  12.  3.  2,  8. 

4.  Flour,  3f^;  sugar,  5^.  6.  57  pear  trees;  43  apple  trees. 

5.  Man,  $3;  boy,  $2.  7.  Silk,  $1.80:  mtiu.  $1.50. 


XXXiv  SCHOOL  ALGEBRA 

9.  Iron,  480  lb.;  lead,  700  lb.  13.  $4600;  $5400. 

10.  First,  3  points;  second,  1  point.  14.  67|  and  172f . 

11.  First,    5    points;    second,    3  15.  50  —  3. 

points;  third,  1  point.  17.  Wheat,  $7;  potatoes,  $50. 

18.  Wheat,  $8;  corn,  $20;  potatoes,  $24. 

19.  First-class,  $1.52;  second-class,  $1.36. 

20.  Fourth-class,  $1.02;  fifth-class,  $.81;  sixth-class,  $.70. 

21.  Corn,  2,772,000,000  bu.;   wheat,  737,000,000  bu.;   oats,  1,007,- 
000,000  bu. 

22.  Copper,  550  lb.;  iron,  480  lb.;  aluminum,  156  lb. 

23.  .2875  in.,  .5025  in.,  .3625  in. 

24.  Eiffel  Tower,  984  ft.;  M.  L.  B'ld'g,  700  ft.;  Wash.  Mon.,  555  ft 

25.  Nitrogen,  15^;  potash,  5^;  phosphate,  5?f. 

26.  Oats,  461  bu.;  corn,  53^  bu. 

27.  20  lb.  of  20j!f  coffee;  40  lb.  of  32^  coffee. 

29.  40  lb.  of  7H  tea;  60  lb.  of  50^  tea. 

30.  Cream,  llf  gal.;  milk,  ^^  gal. 

31.  $3250  at  4%;  $1800  at  5%. 

32.  $2000  at  5%;  $10,000  at  4%. 

34.  /'  X  5''. 

35.  15'  X  6'. 

36.  12  boys;  $60. 

37.  90  mi. 

39.  13  played,  8  won. 

40.  68  cases,  50  successful. 

41.  f. 

42.  -i^. 

52.  A,  in  24  da.;  B,  in  48  da. 

53.  A,  14 A  da.;  B,  18tV  da.;  C,  34f  da. 
57.  49.  58.  23.  59.  64.  60.  151. 

61.  Oarsman,  4  mi.;  stream,  2  mi.  per  hr. 

62.  Oarsman,  5^  mi.;  stream,  If  mi.  per  hr. 

63.  Oarsman,  6  mi.;  stream,  1^  mi.  per  hr. 

64.  Cast  iron,  450  lb.;  wrought  iron,  480  lb. 

65.  From  earth,  93,000,000  mi.;  from  Mars,  141,000,000  mi. 


44.  x\. 

45.  ¥.  i. 

46.  1. 

47.  16,  81. 

48.  21,  79. 

49.  14,  54. 

50.  32,  18. 

51.  51  hr.; 

17  hr. 

ANSWERS  XXXV 

66.  Tea,  50ff;  coffee,  30^. 

67.  24  bu.  from  1st;  16  bu.  from  2d. 

68.  A,  $70;  B,  $110. 

69.  80  lb.  of  25f^  spice;  120  lb.  of  50ff  spice. 
70.  480  mi.  71.  11,  36. 

'     2*2*  '  p-q'  q  -p  * 

EXERCISE  77 

11.  (1)  5;     (2)  V45;     (3)  13;     (4)  Vu. 
12.   25  sq.  spaces.  13.  42  sq.  spaces.  14.   171  ftsj.  spaces. 


1,0. 


EXERCISE  79 

1.  2,  1. 

3.   -  4,  -  2.           5.   -  4,  -  1. 

7.  - 

2.  1,  -  1. 

4.  0,  0.                    6.  2,  3. 

8.   - 

9.  f.                         11.  2.9  -h, 

-3.3-. 

EXERCISE   80 

1.  60  mi.;  1:30  p.  m.  6.  3:7, 

3.  251f  mi.;  5:17^p.  M.  8.2:1. 

5.  17: 10.  10.  At  end  of  5  hr.  20  mi.  from  1*. 

EXERCISE  81 

4.  tV  q    a'+x^  16.  7,  10. 

5.^.  '      "^     '  17.-^,^. 

J  10,-2,  a-b   a  +  b 

^•^^Ay'  11.   -f.  18.3,2,1. 

12.  -  3,  19.  I,  f . 

13.  4.  __      a' -a        h'-b 
^0. 


x{x-] 

ly 

3(4x- 

15) 

5(2x- 

-3) 

x^  +  l 

x^-l 

23.  a' 

-bK 

25.  '-^ 

-ad 

14    8    i  '  a'b—ab'    ab' —a'b 

15.  ii  i.  22.   -  a  -  1. 

27.  11,  9,  18. 

28.  9^(4x2 +  2xy +  2/2). 
a  —  c                                      30.  c  —  a  +  b,  — a  —  6  —  c. 


XXXvi  SCHOOL  ALGEBRA 

32-  7H.  48.  a  =  1^. 

33.  64.  2M 

36.  4,  3.  _  ar"  —a 

42.  174  -  °  F.;  79  -  °  C.  '^"'  ^  "    ^  _  1  * 


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